\subsection{\@startsection {subsection}{1}{\z@}%

\newtheorem{theorem}{Theorem}[section]
\newtheorem*{kemperman}{Kemperman's Theorem}
\newtheorem*{kneser}{Kneser's Theorem}
\newtheorem*{knesers}{Kneser's Density Theorem}
\newtheorem*{folner}{F\o lner's Theorem}
\newtheorem*{griesmer}{Griesmer's Theorem}
\newtheorem*{rigidity}{Rigidity of critical pairs}
\newtheorem*{donghoonlee}{Dong Hoon Lee's Theorem}
\newtheorem*{relative}{Relativization of critical pairs}
\newtheorem*{kriz}{Kriz' example}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{lemma}{Lemma}[section]
\newtheorem*{correspondence}{Correspondence Principle}
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\theoremstyle{definition}\newtheorem{remarks}[theorem]{Remarks}
\theoremstyle{definition}\newtheorem*{acknowledgments}{Acknowledgments}

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\begin{document}

\title{Small product sets in compact groups}

\author{Michael Bj\"orklund}
\address{Department of Mathematics , ETH Z\"urich, Z\"urich, Switzerland}
\curraddr{Department of Mathematics, Chalmers, Gothenburg, Sweden}
\email{micbjo@chalmers.se}
\thanks{}

\subjclass[2010]{Primary: 11P70 ; Secondary: 22C05}

\keywords{Product set theorems, inverse theorems, compact groups}

\date{}

\dedicatory{}

\begin{abstract}
This paper addresses the structure of pairs $(A,B)$ of Borel sets in a compact and second countable group $G$ with Haar probability measure $m_G$
which have positive $m_G$-measures and satisfy the identity
\[
m_G(AB) = m_G(A) + m_G(B) < 1.
\]
When $G$ is a compact, \emph{abelian and connected} group, M. Kneser
provided a very satisfactory description of such pairs, which
roughly says that both $A$ and $B$ must be co-null subsets of pre-images
of two closed intervals under the same surjective homomorphism onto
the circle group. Much more recently, Griesmer was able to divide the
set of all possible such pairs in any compact and \emph{abelian} group
into four (not completely disjoint) subclasses.

Motivated by some applications to small product sets in \emph{countable
amenable groups} with respect to the left upper Banach density, we continue this study, and offer a description of  pairs of Borel sets as above in any
compact and second countable group with an \emph{abelian} identity
component. The relevance of these compact groups to the study of product
sets in countable amenable groups will be explained. The main result of
this paper asserts that in this \emph{non-abelian} situation, essentially only one new class of examples of pairs as above can occur.

Unfortunately, due to a wide variety of technicalities we will have to spend a non-trivial part of the introduction to carefully set up the tools and concepts necessary for the formulation of the main results and to prepare for their proofs.
\end{abstract}

\maketitle

\section{Introduction}

\subsection{General comments}
Let $G$ be a compact and second countable group with Haar probability
measure $m_G$. Suppose $(A,B)$ is a pair of Borel sets in $G$ and define
their \emph{product set} by
\[
AB = \Big\{ ab \, : \, a \in A, \: b \in B \Big\}.
\]
Although the product set of $(A,B)$ may fail to be Borel measurable, it is always measurable with respect to the \emph{completion} of the Borel
$\sigma$-algebra relative to $m_G$, so in particular the Haar measure
of the product set $AB$ is always well-defined.

A well-studied and fundamental question in additive combinatorics is how "small" the $m_G$-measure of $AB$ can be in terms of $m_G(A)$ and
$m_G(B)$. Since $m_G$ is invariant under both left and right translations,
it is clear that one always has the lower bound
\[
m_G(AB) \geq \max(m_G(A),m_G(B))
\]
and equality happens if and only if either $A \subset B$ and $B$ is a
conull subset of an open subgroup of $G$ or $B \subset A$ and
$A$ is a conull subset of an open subgroup of $G$.

A less extreme example of a "small" product set would be when
\begin{equation}
\label{sub-critical1}
m_G(AB) < \min(1,m_G(A) + m_G(B)).
\end{equation}
This kind of pairs have been extensively studied over the years, and are
often referred to as \emph{sub-critical} pairs, although sometimes the
term \emph{critical} is also used, which may cause some confusion when
comparing the results in this paper with other results in the literature.

When $G$ is \emph{abelian}, Kneser \cite{K56} showed that if $(A,B)$ is a
sub-critical pair, then there exist a
\emph{finite} abelian group $M$, a homomorphism $p : G \rightarrow M$
and subsets $I, J \subset M$ such that $m_G(AB) = m_M(IJ)$, where
$m_M$ is the normalized counting measure (Haar measure) on $M$,
and
\[
A \subset p^{-1}(I) \quad \textrm{and} \quad B \subset p^{-1}(J)
\]
and
\begin{equation}
\label{overshoot}
m_M(IJ) = m_M(I) + m_M(J) - m_M(\{e\}),
\end{equation}
where $e$ denotes the identity element in $M$. In other words, the
problem of understanding the structure of Borel sets $(A,B)$ which
satisfy \eqref{sub-critical1} "reduces" to a large extent to the problem
of understanding the structure of subsets $(I,J)$ in a \emph{finite}
abelian group $M$ which satisfy \eqref{overshoot}. In the case when
$G$ is finite cyclic group of prime order, the latter problem was solved
by Vosper in \cite{Vo}. The situation for general finite \emph{abelian}
groups was later outlined by Kemperman in \cite{K60}.

When $G$ is \emph{not} necessarily abelian, Kemperman showed in \cite{K64} that Kneser's "reduction" essentially prevails, modulo the
identity \eqref{overshoot}, that is to say, if $(A,B)$ is a sub-critical
pair in $G$, then there exists a \emph{finite} group $M$,
a homomorphism $p : G \rightarrow M$ and subsets $I, J \subset M$ with
$m_G(AB) = m_M(IJ)$ such that
\[
A \subset p^{-1}(I) \quad \textrm{and} \quad B \subset p^{-1}(J),
\]
but the identity \eqref{overshoot} may fail. Recently, DeVos \cite{DV}
was able to find a reasonable substitute for
\eqref{overshoot} and prove some partial results
on the structure of the sets which satisfy this constraint. \\

Since compact and \emph{connected} groups do not admit any non
trivial homomorphisms onto finite groups, Kemperman's Theorem
(and Kneser's Theorem in the case of abelian groups) in particular
shows that if $(A,B)$ is a pair of Borel sets in such a group, then
\begin{equation}
\label{kneser bound1}
m_G(AB) \geq \min(1,m_G(A) + m_G(B)).
\end{equation}
Clearly, if $G$ is the circle group and $A$ and $B$ are co-null
Borel sets of two closed intervals in $G$, then $(A,B)$ satisfies
\begin{equation}
\label{critical1}
m_G(AB) = \min(1,m_G(A) + m_G(B)),
\end{equation}
which shows that the inequality \eqref{kneser bound1} is sharp
in this case. We shall refer to Borel pairs $(A,B)$ in a compact
group $G$ which satisfies \eqref{critical1} as \emph{critical pairs}
(but we again stress that this is not a completely standard notion,
and the reader should be cautious when comparing the results in
this paper with other results in the literature).

It is not hard to see that if $G$ is a compact group
which admits a surjective homomorphism $p$ onto the circle
group $\mathbb{T}$ and $(I,J)$ is a pair of closed intervals in $\mathbb{T}$
and $A$ and $B$ are co-null subsets of $p^{-1}(I)$ and
$p^{-1}(J)$ respectively, then $(A,B)$ is a critical pair, i.e. it satisfies
\eqref{critical1}.

When $G$ is a \emph{compact and connected abelian group},
Kneser established a beautiful converse of this observation,
namely that the existence of a pair of Borel sets in $G$ which
satisfies \eqref{critical1} \emph{forces} the existence of a
homomorphism $p : G \rightarrow \mathbb{T}$ and closed intervals
$I, J \subset \mathbb{T}$ as above. The situation for general
compact (not necessarily abelian) connected groups
seems to be largely unknown. \\

In order to establish his theorem on critical pairs in compact
and connected \emph{abelian} groups, Kneser utilized a
very useful tool in additive combinatorics known as the
\emph{Dyson $e$-transform}. This "transform" roughly
works as follows. Let $G$ be a compact \emph{abelian}
group and suppose $(A,B)$ is a pair of Borel sets in $G$
with positive $m_G$-measures. Set $A_o = A$ and $B_o = B$
and for every $n \geq 1$, we choose an element
$x_n \in A_{n-1}$ and define the Borel sets
\[
A_n = A_{n-1} \cup B_{n-1}x_{n} \quad \textrm{and} \quad B_n = x_n^{-1}A_{n-1} \cap B_{n-1}.
\]
Since $G$ is abelian, we have
\[
A_n \subset A_{n+1} \quad \textrm{and} \quad B_n \supset B_{n+1} \
\]
and
\[
A_n B_n \subset AB \quad \textrm{and} \quad m_G(A_n) + m_G(B_n) = m_G(A) + m_G(B)
\]
for all $n$. In particular, if $(A,B)$ is a critical pair in $G$, then
$(A_n,B_n)$ is either a critical or sub-critical pair for every $n$, and
upon studying the "limits" of these pairs, one can hope to deduce some structural
information about the pair $(A,B)$. Kneser assumed
connectedness of $G$ in order to avoid some technical issues
with open subgroups in the general setting. More recently,
Griesmer in \cite{G11} (inspired by another recent paper \cite{Gr09}
by Grynkiewicz) was able to carry out the full analysis of
critical pairs in compact \emph{abelian} groups using the Dyson
$e$-transform and to conclude that one could divide the possible
critical pairs in any compact \emph{abelian} into four (not disjoint)
sub-classes.

According to this division, a critical pair $(A,B)$ of Borel sets in $G$ is
either (up to some translations) of the form as in Kneser's Theorem, i.e. the sets $A$ and $B$ are conull subsets of
pre-images of closed intervals under a homomorphism of an open subgroup of $G$ to the circle group \emph{or} there exists an open subgroup
$U < G$ and conull Borel sets $A' \subset A$ and $B' \subset B$ such
that for some $x,y \in G/U$, the pair $(x^{-1}A \cap U,By^{-1} \cap U)$ is sub-critical in $U$. We shall refer to the latter situation by saying that
$(A,B)$ is a \emph{locally sub-critical pair} in $G$. Griesmer \cite{G11}
has been able
to identity three (not mutally different) ways in which this local sub-criticality takes place. We shall not reproduce his findings here.

\subsection{A relativisation of Kneser's Theorem}
The aim of this paper is to understand the structure of critical Borel
pairs in compact, but not necessarily \emph{abelian} groups. Because
of the possible non-abelianess of the groups under study, we cannot
hope to succesfully employ Dyson's $e$-transform as in Kneser's and
Griesmer's approaches, so we have to develop an alternative route, which
we shall now briefly outline.

Let $G$ be a compact and second countable group and let $N$ be a
closed \emph{normal} subgroup of $G$. We may view the Haar probability
measure on $N$ as the unique $N$-invariant probability measure
supported on $N < G$. Similarily, given any element $x \in G/N$, the
right translate of $m_N$ by $x$ is a $N$-invariant probability measure
on the coset $Nx$, and we use the rather ugly symbol $(m_G)_x^N$
to denote this probability measure. One readily checks that the map
$x \mapsto (m_G)_x^N$ from $G/N$ to the space of Borel probability
measures on $G$ is weak*-continuous and by the uniqueness of
$m_G$, we have the following disintegration of $m_G$,
\begin{equation}
\label{disintegration1}
m_G = \int_{G/N} \big(m_G\big)_x^N \, dm_{G/N}(x),
\end{equation}
where $m_{G/N}$ denotes the push-forward of $m_G$ onto the
quotient group $G/N$.

Suppose that $(A,B)$ is a critical Borel pair in $G$ and let $N$ be a
closed normal subgroup. For every $x, y \in G$, we can
define the Borel sets
\[
A_x = x^{-1}A \cap N \quad \textrm{and} \quad B^y = By^{-1} \cap N
\]
in $N$. Note that since $N$ is assumed to be normal, we have
$A_x B^y \subset x^{-1}\big(AB \cap Nxy\big)y^{-1}$ and
\[
\big(m_G\big)_x^N(A) = m_N(x^{-1}A)
\quad \textrm{and} \quad
\big(m_G\big)_y^N(B) = m_N(By^{-1}).
\]
We shall say that a Borel pair $(A,B)$ is \emph{critical with respect to $N$}
if there exist conull Borel subsets $X \subset G$ and $Z \subset X \times X$
such that
\[
\big(m_G\big)_x^N(A) = m_G(A) \quad \textrm{and} \quad
\big(m_G\big)_y^N(B) = m_G(B)
\]
for all $x,y \in X$ and
\[
\big(m_G\big)_{xy}^N(AB)  = \big(m_G\big)_x^N(A) + \big(m_G\big)_y^N(B)
\]
for all $(x,y) \in Z$. \\

Note that if $N$ is \emph{connected} and $(A,B)$ is a
critical pair with respect to $N$, then all of the pairs $(A_x,B^y)$ are critical for $(x,y) \in Z$, that is to say, in this case we have a function from $Z$ into the set of critical pairs in $N$. Hence, if $N$ in addition is \emph{abelian},
then Kneser's Theorem can be applied to each of these pairs, producing for
every $z$ in $Z$ a surjective homomorphism $\pi_z$ of $N$ onto the circle group $\mathbb{T}$ and two elements $\alpha_z, \beta_z \in \mathbb{T}$ such that
\[
A_x \subset \pi_z^{-1}(\alpha_z I) \quad \textrm{and} \quad B^y \subset \pi_z^{-1}(J\beta_z),
\]
where $I$ and $J$ are the uniquely determined closed intervals in $\mathbb{T}$
which are symmetric around zero and have normalized arc lengths (Haar
measures) equal to $m_G(A)$ and $m_G(B)$ respectively. \\

A prominent class of examples of critical pairs with respect to a closed normal subgroup  can be given as follows. Let $G$ be a
compact group which is a semidirect product of the form
$G = N \rtimes K$, where $N$ and $K$ are closed subgroups and
$K$ acts on $N$ by conjugation, and suppose that $\pi$ is a continous homomorphism onto either $\mathbb{T}$ or the "twisted" torus
$\widetilde{\mathbb{T}} = \mathbb{T} \rtimes \{-1,1\}$. In the first case, it is not
hard to see that the pair $(\pi^{-1}(I),\pi^{-1}(J))$ is critical with respect
to $N$ for \emph{any} pair of closed intervals $I, J \subset \mathbb{T}$.

In the second case, we let $I$ and $J$ be closed and \emph{symmetric} intervals in $\mathbb{T}$ and define the closed subsets $\widetilde{I} = I \rtimes \{-1,1\}$ and $\widetilde{J} = J \rtimes \{-1,1\}$ of $\widetilde{T}$. One can check that $(\widetilde{I}, \widetilde{J})$ is a critical pair in $\widetilde{T}$ and that the pair
\[
A = \pi^{-1}(\widetilde{I})
\quad \textrm{and} \quad
B = \pi^{-1}(\widetilde{J})
\]
is critical in $G$ with respect to $N$. We shall refer to this class of examples
as \emph{sturmian pairs}.

The main novelty in this paper is the observation that if $G$ is any compact and second countable group with a closed and connected subgroup $N$ such that $G$ can be written on the semi-direct form $G = N \rtimes K$ (one often says that $N$ \emph{splits} in $G$) and if $(A,B)$ is a Borel pair in $G$ which is critical with respect to $N$, then the map $(x,y) \mapsto (A_x,B^y)$ into the set of critical pairs in $N$ must essentially come from a sturmian pair.

Recall that the \emph{identity component} of a compact group $G$ is
defined as the intersection of all \emph{open} subgroups and is thus
automatically normal (in fact, it is fully characteristic, i.e. invariant under
all comtinuous automorphisms of $G$). If $G$ is also second countable,
then the identity component can be realized as the intersection of a \emph{countable} number of open (normal) subgroups.

\begin{theorem}
\label{thm1}
Let $G$ be a compact and second countable group with a \emph{split}
identity component $N$ and suppose $(A,B)$ is a critical pair in $G$
with respect to $N$. Let $I$ and $J$ denote the closed and symmetric
intervals in $\mathbb{T}$ whose Haar probability measures equal $m_G(A)$ and $m_G(B)$ respectively. If $N$ is abelian, then either there exists a
continuous homomorphism $\pi_o : G \rightarrow \mathbb{T}$ such that
\[
A \subset \pi_o^{-1}(s'I)
\quad \textrm{and} \quad
B \subset \pi_o^{-1}(Jt')
\]
for some $s', t' \in \mathbb{T}$, or there exists a continuous
homomorphism $\pi : G \rightarrow \mathbb{T} \rtimes \{-1,1\}$ such that
\[
A \subset \pi^{-1}\Big(s' \, \big(I \rtimes \{-1,1\}\big)\Big)
\quad \textrm{and} \quad
B \subset \pi^{-1}\Big(\big(J \rtimes \{-1,1\}\big) \, t'\Big)
\]
for some $s', t' \in \mathbb{T} \rtimes \{-1,1\}$.
\end{theorem}

Unfortunately, there are (quite intricate) examples of compact and second
countable groups whose identity components do \emph{not} split as
above into a semi-direct product, so Theorem \ref{thm1} is not
directly applicable to those groups. However, the following theorem of
Lee \cite{DHL} allows us to at least partially circumvent this technicality.

\begin{donghoonlee}
Let $G$ be a compact group and let $N$ denote its identity component.
Then there exists a closed totally disconnected subgroup $K$ of $G$ and a
surjective homomorphism $p : N \rtimes K \rightarrow G$ whose kernel is
contained in the center of $N \cap K$.
\end{donghoonlee}

In other words, if $(A,B)$ is a critical pair in a compact group $G$ with
identity component $N$, then $(p^{-1}(A),p^{-1}(B))$ is a critical pair
in the semi-direct product $N \rtimes K$, for some totally disconnected
subgroup $K$ of $N$ and surjective homomorphism
$p : N \rtimes K \rightarrow G$.\\

Theorem \ref{thm1} settles the question about the structure of those
critical pairs which are critical \emph{with respect to an abelian
identity component}. However, far from every critical pair in a compact
(non-connected) group is critical with respect to its identity component.

The second main observation of this paper roughly says that if a critical
pair in a compact and second countable group $G$ is \emph{not} critical
with respect to the identity component of the group, then the pair must
exhibit substantial "periodicity"; in fact,
such a Borel pair $(A,B)$ in $G$ must be \emph{locally sub-critical}, i.e. there exists an open normal subgroup $U < G$ and $x, y \in G$ such that the pair
$(x^{-1}A \cap U,By^{-1} \cap U)$ is sub-critical in $U$. Formally, this
"relativization" can be stated as follows.

\begin{relative}[Appendix I]
Let $G$ be a compact and second countable group and suppose $(A,B)$ is a critical pair in $G$ which is not locally sub-critical.  Then there exist
an open subgroup $L < G$ and conull $\sigma$-compact sets
\[
A' \subset A \quad \textrm{and} \quad B' \subset B
\]
and $s,t \in G$ such that
\[
s \, A' \subset L \quad \textrm{and} \quad B' \, t \subset L
\]
and $(s \, A', B' \, t)$ is critical in $L$ with respect to the identity component of $L$.
\end{relative}

This relativization principle, together with Theorem \ref{thm1} and Dong-Hoon Lee's Theorem, shows that a critical pair $(A,B)$ which are not sub-critical pairs, admits \emph{conull} subsets $A' \subset A$ and
$B' \subset B$ with a rather satisfactory structure theory, namely they
are realized as conull subsets of sturmian pairs as above. However, a
priori, the original sets $A$ and $B$ do not need to be contained in
these sturmian pairs. In the next subsection, we shall discuss a quite
general "rigidity" principle which utilizes the special topological structure
of sturmian pairs to conclude that one indeed does have global containment
in a sturmian pair.

\subsection{Reduction theory of critical pairs}
At this point we have seen how a series of operations can be applied to a
critical pair in order to put it in a form so that Theorem \ref{thm1} can be
applied. It makes sense to introduce a concise notion which neatly summarizes these operations. \\

In order to motivate this notion, we first observe that we so far have used
three different ways to construct "new" critical pairs from a given critical
pair $(A,B)$ in a compact group $G$. Firstly, note that if $L$ is an open
(not necessarily normal) subgroup of $G$ and there exists $s, t \in G$
such that
\[
sA \subset L \quad \textrm{and} \quad Bt \subset L,
\]
then $(sA,Bt)$ is a critical pair in $L$. Secondly, if $S$ is a compact group
and $p : S \rightarrow G$ is a continuous surjective homomorphism, then
\[
\big(p^{-1}(A),p^{-1}(B)\big)
\]
is critical in $S$. Thirdly, if it so happens that there exists a compact subgroup $K < \operatorname{Aut}(G)$ which fixes the set $B$, then
\[
\big(A \rtimes K, B \rtimes K\big)
\]
is a critical pair in the semi-direct product $S = G \rtimes K$. Upon
combining these three ways to construct critical pairs, we arrive at
the following "reduction theory". \\

Given two Borel sets $A, A' \subset G$, we shall say that
$A$ is \emph{almost contained} in $A'$ and denote this
by $A \subsetsim A'$ if the relative complement $A \setminus A'$
is a null set in $G$.

\begin{definition}
Let $G$ and $M$ be compact groups and suppose $(A,B)$ and $(I,J)$ are
two pairs of Borel sets of $G$ and $M$ respectively. We shall say that
$(A,B)$ \emph{almost reduces} to $(I,J)$, and write
\[
(A,B) \preceq_a (I,J),
\]
if the identity $m_G(AB) = m_M(I  J)$ holds
and there exist an open (not necessarily normal) subgroup $L < G$ and $s, t \in G$ such that $s \, A \subset L$ and $B \, t \subset L$
and a compact group $S$, a continuous
homomorphism $p : S \rightarrow L$ and a
compact (possibly trivial) subgroup $K < \operatorname{Aut}(M)$, which fixes
the set $J$ such that
\[
p^{-1}(s \, A) \subsetsim \pi^{-1}\Big(s' \big(I \rtimes K\big)\Big)
\quad \textrm{and} \quad
p^{-1}(B \, t) \subsetsim \pi^{-1}\Big(\big(J \rtimes K\big) \, t' \Big)
\]
for some $s', t' \in M \rtimes K$. We shall say that $(A,B)$
\emph{reduces} to $(I,J)$, and write
\[
(A,B) \preceq (I,J),
\]
if the almost containments above are inclusions, and we say that $(A,B)$ \emph{strictly reduces} to  $(I,J)$ if we have
\[
G = S = L
\]
that is to say, if no passage to neither subgroups nor supergroups is
necessary.
\end{definition}

\subsection{Rigidity of critical pairs}

From the discussion in the last subsection, we observe that Theorem
\ref{thm1}, together with Dong-Hoon Lee's Theorem and the Relativization
Principle, implies that if $G$ is a compact and second countable group
with an abelian identity component, and $(A,B)$ is a critical pair in $G$
which is not locally sub-critical, then $(A,B)$ \emph{almost reduces} to
a pair of closed and symmetric intervals $(I,J)$ in $M = \mathbb{T}$. We now
wish to show that because of the special topological structure of closed
intervals in $\mathbb{T}$, namely that they are equal to the closures of their
interiors and the identities
\begin{equation}
\label{stable}
I = \bigcap_{j \in J} IJj^{-1} \quad \textrm{and} \quad J = \bigcap_{i \in I} i^{-1}IJj
\end{equation}
hold, the almost reduction is in fact a reduction. \\

The general framework for this "rigidity" of critical pairs can be described as follows. Let $M$ be a compact group. We say that a Borel set $I \subset M$ is \emph{regular}
if it equals the closure of its interior, and we say  that a pair $(I,J)$ of Borel sets in $M$ is \emph{left stable} if the inclusion
\[
x \, J \subset I \, J
\]
implies that $x \in I$. There is an obvious analogous notion of right
stability, and we say that $(I,J)$ is \emph{stable} if it is both left and
right stable. One readily checks that pairs of intervals in $\mathbb{T}$ are
stable, that is, the identities in \eqref{stable} hold, and clearly closed
intervals in $\mathbb{T}$ are regular.

\begin{rigidity}[Appendix II]
Let $G$ and $M$ be compact groups and suppose $(A,B)$ and $(I,J)$ are critical pairs in $G$ and $M$ respectively. If $(I,J)$ is stable and regular
and if $(A,B)$ almost reduces to $(I,J)$, then $(A,B)$ in fact reduces to
$(I,J)$.
\end{rigidity}

We can now neatly summarize the above discussions in the following
theorem, which is a structural result for arbitrary critical pairs in any
compact and second countable group with an abelian identity component.

\begin{theorem}
\label{cor1}
Let $G$ be a compact and second countable group with an abelian
identity component. If $(A,B)$ is a critical pair in $G$, which is
not locally sub-critical, then $(A,B)$ reduces to a sturmian pair.
\end{theorem}

In Appendix III, we shall discuss some applications of Theorem \ref{thm1}
and Theorem \ref{cor1} to problems about product sets in countable
amenable groups, and hopefully also clarify the relevance of compact
groups with abelian identity components in this setting.

\begin{remark}
It is not hard to show that if $G$ is abelian and $(A,B)$ is a critical
pair in $G$, then we can in fact find an open subgroup $L < G$, a
continuous surjective homomorphism $\theta : L \rightarrow \mathbb{T}$ and closed intervals $I, J \subset \mathbb{T}$ with
\[
m_L(A) = m_{\mathbb{T}}(I) \quad \textrm{and} \quad m_L(B) = m_{\mathbb{T}}(J)
\]
such that $s \, A \subset L$ and $B \, t \subset L$ and
\[
s \, A \subset \theta^{-1}(I) \quad \textrm{and} \quad B \, t \subset \theta^{-1}(J)
\]
for some $s, t \in G$, that is to say, in the case when $G$ is abelian,
the passage from $L$ to a supergroup $S$ in the definition of a
reduced pair is unnecessary.
\end{remark}

\subsection{An outline of Theorem \ref{thm1}}

The aim of this section is to reduce the proof of Theorem \ref{thm1}
to two propositions, of which the first one constitutes the technical
core of this paper and the second one follows rather easily from
standard arguments.

We shall say that a compact and connected group $N$ is a Kneser group if
every critical pair in $N$ \emph{strictly reduces} to a
sturmian pair, and if $G$ is a compact group with identity component $N$, then we say that $N$ \emph{splits} if there exists a totally disconnected subgroup $Q < G$ such that $G$ is isomorphic (as a topological group)
to the semi-direct product $G = N \rtimes Q$.

\begin{proposition}
\label{prop1}
Let $G$ be a compact and second countable group with a split Kneser identity component $N$. If $(A,B)$ is a critical pair in $G$ with respect to $N$, then there exist a continuous surjective homomorphism $\xi : N \rightarrow \mathbb{T}$ and closed symmetric intervals $I_o, J_o \subset \mathbb{T}$ with
\[
m_{G}(A) = m_{\mathbb{T}}(I_o) \quad \textrm{and} \quad m_{G}(B) = m_{\mathbb{T}}(J_o)
\]
and conull Borel sets $X \subset G$ and $Y \subset X \times X$
and two Borel measurable maps $\alpha_o, \beta_o : X \rightarrow \mathbb{T}$
such that
\[
p^{-1}A \cap N \subset \xi^{-1}\Big(I_o \, \alpha_o(p)\Big)
\quad \textrm{and} \quad
Bq^{-1} \cap N \subset \xi^{-1}\Big(J_o \, \beta_o(q)\Big)
\]
for all $(p,q) \in Y$. Furthermore, there exists a continuous homomorphism
$\epsilon : G/N \rightarrow \{-1,1\}$ such that
\[
\xi(c_{p}(n)) = \xi(n)^{\epsilon(n)}, \quad \forall \, (n,p) \in N \times G/N
\]
and
\[
\Big(\alpha_o(p) \beta_o(q) \Big)^{\epsilon(p)}
=
\Big(\alpha_o(p') \beta_o(q') \Big)^{\epsilon(p')},
\]
whenever $(p,q)$ and $(p',q')$ belong to $Y$
and satisfy $pq = p'q'$, where $c_p$ denotes
the automorphism of $N$ induced by $p$ in $G$.
\end{proposition}

The strange-looking identities at the end of the formulation of Proposition
\ref{prop1} will be used to build Borel measurable maps from a conull Borel
subset $X \subset G$ to either the one-dimensional torus $\mathbb{T}$ or the "twisted" torus
$\mathbb{T} \rtimes \{-1,1\}$ with the property that there exist a conull Borel
set $Z \subset X \times X$ such that
\[
\alpha(x) \, \beta(y) = \alpha(x') \, \beta(y')
\]
for all $(x,y) \in Z$ with $xy = x'y'$. The following propositon (which follows from rather standard arguments) shows that
this kind of maps must come from translations of continuous homomorphisms between the relevant groups.

\begin{proposition}
\label{prop2}
Let $G$ and $M$ be compact and second countable groups and suppose
there exist conull Borel sets $X \subset G$ and $Z \subset X \times X$
and two Borel measurable maps $\alpha, \beta : X \rightarrow M$ such that
\[
\alpha(x) \beta(y) = \alpha(x') \beta(y')
\]
whenever $(x,y)$ and $(x',y')$ belong to $Z$ and
satisfy $xy = x'y'$. Then there exists a continuous homomorphism
$\pi : G \rightarrow M$ and $s, t \in M$ such that
\[
\alpha(x) = s \, \pi(x) \quad \textrm{and} \quad \beta(y) = \pi(y) \, t
\]
for almost every $(x,y)$.
\end{proposition}

\begin{proof}
By Fubini's Theorem, there exist $x_1, x_2 \in  X$ such that
the sets
\[
Z^{x_1} = \Big\{ x \in G \, : \, (x_1,x) \in Z \Big\}
\quad \textrm{and} \quad
Z_{x_2} = \Big\{ x \in G \, : \, (x,x_2) \in Z \Big\}
\]
are conull Borel sets in $G$. Note that
\[
X' = x_1^{-1} \, X \cap X\, x_2^{-1}
\quad \textrm{and} \quad
Z' = (x_1,e)^{-1} \, Z \, (e,x_2)^{-1} \cap \Big( X_o \times X_o\Big)
\]
are again conull Borel sets in $G$ and $G \times G$ respectively and
\[
Z'^{e} = Z^{x_1} \, x_2^{-1} \quad \textrm{and} \quad Z'_e = x_1^{-1} \, Z_{x_2}.
\]
If we set
\[
X'' = Z'^e \cap Z'_e \quad \textrm{and} \quad Z'' = Z' \cap p^{-1}(X'') \cap (X'' \times X''),
\]
where $p : G \times G \rightarrow G$ denotes the product map, then $X''$ and
$Z''$ are conull subsets,
\[
e \in X'' \quad \textrm{and} \quad (e,e) \in Z''
\]
and the functions
\[
\gamma_1(x) = \alpha(x_1)^{-1} \, \alpha(x_1 \, x)
\quad \textrm{and} \quad
\gamma_2(x) = \beta(x \, x_2) \, \beta(x_2)^{-1}
\]
are Borel measurable on $X''$ and satisfy
$\gamma_1(e) = \gamma_2(e) = e$ and
\[
\gamma_1(x) \, \gamma_2(y) = \gamma_1(x') \,  \gamma_2(y')
\]
whenever $(x,y)$ and $(x',y')$ belong to $Z''$ and $xy = x'y'$. In
particular, $\gamma_1(x) = \gamma_2(x)$ for all $x \in X''$ and
the equation
\[
\gamma_1(xy) = \gamma_1(x) \, \gamma_1(y)
\]
holds whenever $(x,y)$ belongs to $Z''$.

By Theorem B.2 in \cite{Zi}, we conclude that here exists a
continuous homomorphism $\pi : G \rightarrow M$ such that
$\pi(x) = \gamma_1(x)$ almost everywhere, and if we
define
\[
s = \alpha(x_1) \quad \textrm{and} \quad t = \beta(x_2),
\]
then
\[
\alpha(x) = s \, \pi(x) \quad \textrm{and} \quad \beta(x) = \pi(x) \, t
\]
for almost every $x$ in $G$.
\end{proof}

\subsubsection{Proof of Theorem \ref{thm1}}
Let $G$ be a compact and second countable group with a split
identity component which we shall assume is a Kneser group, i.e.
we write
\[
G = N \rtimes Q,
\]
where $Q < \operatorname{Aut}(N)$ and every critical pair in $N$ strictly reduces to
a sturmian pair. Denote by $c_p$ the automorphism of $N$ given by
$p$ in $Q$ and note that the map $p \mapsto c_p$ is assumed to be a continuous homomorphism.

The group multiplication in $G$ is given by
\[
(m,p) (n,q) = (m \, c_p(n), pq), \quad \forall \, (m,p), (n,q) \in G.
\]
In particular, the equation
\[
(m,p)(n,q) = (m',p')(n',q')
\]
holds if and only if
\[
m \, c_p(n) = m' \, c_{p'}(n) \quad \textrm{and} \quad pq = p'q'.
\]
Suppose $(A,B)$ is a critical pair in $G$ with respect to $N$. By
Proposition \ref{prop1}, there exist a continuous surjective
homomorphism $\xi : G \rightarrow  \mathbb{T}$, closed symmetric intervals
$I_o, J_o \subset \mathbb{T}$ with
\[
m_G(A) = m_{\mathbb{T}}(I_o) \quad \textrm{and} \quad m_G(B) = m_{\mathbb{T}}(J_o)
\]
and conull Borel sets $X \subset G$ and $Y \subset X \times X$
and two Borel measurable maps $\alpha_o, \beta_o : X \rightarrow \mathbb{T}$
with the property that
\[
p^{-1}A \cap N \subset \xi^{-1}\Big(I_o \, \alpha_o(p)\Big)
\quad \textrm{and} \quad
B q^{-1} \cap N \subset \xi^{-1}\Big(J_o \, \beta_o(q)\Big)
\]
for all $p, q \in X$ and a continuous homomorphism
$\epsilon : Q \rightarrow \{-1,1\}$ such that
\[
\xi(c_p(n)) = \xi(n)^{\epsilon(n)}, \quad \forall \, (n,p) \in G
\]
and
\[
\Big(\alpha_o(p) \beta_o(q) \Big)^{\epsilon(p)}
=
\Big(\alpha_o(p') \beta_o(q') \Big)^{\epsilon(p')},
\]
whenever $(p,q)$ and $(p',q')$ belong to $Y$
and satisfy $pq = p'q'$. \\

First assume that $\epsilon$ is trivial and define the maps
$\alpha, \beta : N \rtimes X \rightarrow \mathbb{T}$
by
\[
\alpha(m,p) = \alpha_o(p)^{-1} \, \xi(m)
\quad \textrm{and} \quad
\beta(n,q) = \beta_o(q)^{-1} \, \xi(n)
\]
and note that if $(m,p), (n,q) \in N \rtimes X$ satisfy
\[
(m,p)(n,q) = (m',p')(n',q'),
\]
then
\begin{eqnarray*}
\alpha(m,p) \, \beta(n,q)
&=&
\big(\alpha_o(p) \, \beta_o(q)\big)^{-1} \, \xi(m) \, \xi(n) \\
&=&
\big(\alpha_o(p) \, \beta_o(q)\big)^{-1} \, \xi(c_p(m)) \, \xi(n) \\
&=&
\big(\alpha_o(p) \, \beta_o(q)\big)^{-1} \, \xi(c_p(m) \, n) \\
&=&
\big(\alpha_o(p') \, \beta_o(q')\big)^{-1} \, \xi(c_{p'}(m') \, n') \\
&=&
\alpha(m',p') \, \beta(n',q').
\end{eqnarray*}
Furthermore, we have
\[
A \cap (N \rtimes X) \subset \alpha^{-1}(I_o)
\quad \textrm{and} \quad
B \cap (N \rtimes X) \subset \beta^{-1}(J_o)
\]
By Proposition \ref{prop2}, there exists a continuous homomorphism
such that
\[
\alpha(m,p) = s \, \pi(m,p) \quad \textrm{and} \quad \beta(n,q) = \pi(n,q) \, t
\]
for some $s, t \in G$ and for almost every $(m,p)$ and $(n,q)$ in $G$,
which implies that
\[
A \cap (N \rtimes X) \subset \pi^{-1}(s' \, I_o)
\quad \textrm{and} \quad
B \cap (N \rtimes X) \subset \pi^{-1}(J_o \, t')
\]
for some $s', t' \in \mathbb{T}$. In particular, this shows that $(A,B)$ almost
reduces to $(I_o,J_o)$, but since $(I_o,J_o)$ clearly is stable and both
$I_o$ and $J_o$ are regular sets, the rigidity of critical pairs asserts
that in fact $A$ and $B$ are contained in the right hand sides above. \\

Now assume that $\epsilon$ is \emph{not} trivial, which automatically means
that it maps onto $\{-1,1\}$.  Define the maps
\[
\alpha, \beta : N \rtimes X \rightarrow \mathbb{T} \rtimes \{-1,1\}
\]
by
\[
\alpha(m,p) = \Big(\alpha_o(p)^{-\epsilon(p)} \, \xi(m), \epsilon(p)\Big)
\quad \textrm{and} \quad
\beta(n,q) = \Big(\beta_o(q)^{-1} \, \xi(n), \epsilon(q)\Big).
\]
Note that if $(m,p), (n,q) \in N \rtimes X$ satisfy
\[
(m,p)(n,q) = (m',p')(n',q'),
\]
then
\begin{eqnarray*}
\alpha(m,p) \, \beta(n,q)
&=&
\Big(
\big(\alpha_o(p) \, \beta_o(q)\big)^{-\epsilon(p)} \, \xi(m) \xi(n)^{\epsilon(p)}, \,
\epsilon(p) \epsilon(q)
\Big) \\
&=&
\Big(
\big(\alpha_o(p) \, \beta_o(q)\big)^{-\epsilon(p)} \, \xi(m \, c_p(n)), \,
\epsilon(p q)
\Big) \\
&=&
\Big(
\big(\alpha_o(p') \, \beta_o(q')\big)^{-\epsilon(p')} \, \xi(m \, c_{p'}(n')), \,
\epsilon(p'q')
\Big) \\
&=&
\alpha(m',p') \, \beta(n',q').
\end{eqnarray*}
By Propositon \ref{prop2}, this shows that $\alpha$ and $\beta$
are indeed left and right translates of a continuous homomorphism
$\pi : G \rightarrow \mathbb{T} \rtimes \{-1,1\} $ respectively. Since
\begin{eqnarray*}
\alpha^{-1}\Big(I_o \rtimes \{-1,1\}\Big) \cap Np
&=&
\Big\{ (m,p) \, : \, \alpha_o(p)^{-\epsilon(p)} \, \xi(m) \in I_o \Big\}  \cap Np \\
&=&
\xi^{-1}\Big(I_o \, \alpha_o(p)^{\epsilon(p)}\Big) \, p \\
&=&
\xi^{-1}\Big(\big(I_o \, \alpha_o(p)\big)^{\epsilon(p)}\Big) \, p \\
&=&
\Big(\xi \circ c_{p^{-1}}\Big)^{-1}\Big(I_o \, \alpha_o(p)\Big) \, p \\
&\supset &
A \cap Np
\end{eqnarray*}
and
\begin{eqnarray*}
\beta^{-1}\Big(J_o \rtimes \{-1,1\}\Big) \cap Nq
&=&
\Big\{ (m,q) \, : \, \beta_o(q)^{-1} \, \xi(m) \in J_o \Big\}  \cap Nq \\
&=&
\xi^{-1}\Big(J_o \, \beta_o(q)\Big) \, q \\
&\supset &
B \cap Nq,
\end{eqnarray*}
we see that
\[
A \cap (N \rtimes X) \subset \alpha^{-1}\Big(I \rtimes \{-1,1\}\Big)
\quad \textrm{and} \quad
B \cap (N \rtimes X) \subset \beta^{-1}\Big(J \rtimes \{-1,1\}\Big),
\]
so we conclude as before (using the rigidity of critical pairs) that
\[
A  \subset \pi^{-1}\Big(s\big(I \rtimes \{-1,1\}\big)\Big)
\quad \textrm{and} \quad
B \subset \pi^{-1}\Big(\big(J \rtimes \{-1,1\}\big)t\Big),
\]
which finishes the proof.

\subsection{Organization of the paper}
The paper consists of Section \ref{proofProp1} and two appendices.
In Section \ref{proofProp1} we give a proof of Proposition \ref{prop1}.
In Appendix I we establish the principle of relativization of critical
pairs which was formulated in the introduction and used in the
passage from Theorem \ref{thm1} to Corollary \ref{cor1}.
In Appendix II we prove the rigidity of critical pairs which was
formulated in the introduction and used in the last step of the
proof of Theorem \ref{thm1}.

\subsection{Acknowledgements}
The present paper is an outgrowth of discussions between Alexander Fish
and the author on a forthcoming joint paper on analogues of Kneser's and Kemperman's Theorems for the upper Banach densities on countable amenable groups.

It is a pleasure to thank Alexander Fish and John Griesmer
for many interesting discussions on the topic of this paper.
The author also wishes to acknowledge financial support from the ETH Fellowship FEL-171-03.
\section{Proof of Proposition \ref{prop1}}
\label{proofProp1}
The aim of this section is to establish Proposition \ref{prop1}. This will
be done by an iterative use of three lemmata which we shall prove below.\\

The first lemma asserts that two sets in a compact group $G$
which are pre-images of nice enough sets in a compact group
$M$ under two surjective homomorphisms from $G$ onto $M$
and only differ up to null sets must in fact be equal and the
corresponding sets must be related via an automorphism of $M$.

If $M$ is a compact group and $I \subset M$ is a subset, then the
\emph{stabilizer group} $M_I$ of $I$ is defined by
\[
M_I = \Big\{ m \in M \, : \, m \, I = I \, m = I \Big\}.
\]
Recall that a set $I \subset M$ is \emph{regular} if it equals the
closure of its interior in $M$.

\begin{lemma}
\label{sim}
Let $G$ and $H$ be compact and second countable groups and suppose $\pi_1$ and $\pi_2$ are continuous surjective
homomorphisms from $G$ onto $H$. If $I_1, I_2 \subset H$
are regular sets with trivial stabilizer groups such that
\[
\pi_1^{-1}(I_1) \sim \pi_2^{-1}(I_2),
\]
then there exists a continuous automorphism $\alpha$
of $H$ such that
\[
\pi_1 = \alpha \circ \pi_2 \quad \textrm{and} \quad I_1 = \alpha(I_2).
\]
In particular, if $H = \mathbb{T}$ and $I_1$ and $I_2$ are closed intervals, then either $\pi_1 = \pi_2$ and $I_1 = I_2$, or
\[
\pi_1(x) = \pi_2(x)^{-1} \quad \textrm{and} \quad I_1 = I_2^{-1}
\]
for all $x$ in $G$.
\end{lemma}

\begin{proof}
Since continuous \emph{surjective} homomorphisms between $G$
and $H$ are necessarily open maps, we have
\[
\pi_1^{-1}(I_1) = \overline{\pi_1^{-1}(I_1^o)}
\quad \textrm{and} \quad
\pi_2^{-1}(I_2) = \overline{\pi_2^{-1}(I_2^o)},
\]
i.e. the pre-images of $I_1$ and $I_2$ under $\pi_1$ and $\pi_2$
are regular sets in $G$. By assumption, the open sets defined by
\[
U_1 = \pi_1^{-1}(I_1^o) \setminus \pi_2^{-1}(I_2)
\quad \textrm{and} \quad
U_2 = \pi_2^{-1}(I_2^o) \setminus \pi_2^{-1}(I_1)
\]
are Haar null and thus empty, which implies that
\[
\pi_1^{-1}(I_1) = \overline{\pi_1^{-1}(I_1^o)} \subset \pi_2^{-1}(I_2)
\quad \textrm{and} \quad
\pi_2^{-1}(I_2) = \overline{\pi_2^{-1}(I_2^o)} \subset \pi_1^{-1}(I_1).
\]
Hence, $\pi_1^{-1}(I_1) = \pi_2^{-1}(I_2)$,
and since $I_1$ and $I_2$ are assumed to have trivial stabilizer
subgroups, we conclude that $\ker \pi_1 = \ker \pi_2$. Indeed,
if $x \in \ker \pi_1$, then
\[
\pi_2(\pi^{-1}(I_1)) = I_2 = \pi_2(\pi_1^{-1}(I_1)) \, \pi_2(x)
\]
which shows that $\pi_2(x)$ fixes $I_2$ and must thus be zero.
Hence, $\ker \pi_2 \subset \ker \pi_1$, and by symmetry we
conclude that $\ker \pi_1 = \ker \pi_2$.

If we denote by $\pi$ the canonical quotient map from $G$ onto the quotient group $H = G/\ker \pi_1$, then, by the universal property of this map, there exist continuous automorphisms $\alpha_1$ and $\alpha_2$ of $H$ such that
\[
\pi_1 = \alpha_1 \circ \pi \quad \textrm{and} \quad \pi_2 = \alpha_2 \circ \pi.
\]
In particular, if we define $\alpha = \alpha_1 \circ \alpha_2^{-1}$, then
$\pi_1 = \alpha \circ \pi_2$,
which finishes the proof. The last assertion immediately follows from the
fact that $\operatorname{Aut}(\mathbb{T}) = \{-1,1\}$.
\end{proof}

Before we state the second lemma, we first note that if $N$ is a connected
group and if $\xi : N \rightarrow \mathbb{T}$ is a non-trivial homomorphism (hence surjective) then
\[
\check{\xi}(n) = \xi(n)^{-1}, \quad n \in N,
\]
is again a non-trivial homomorphism from $N$ onto $\mathbb{T}$.

\begin{lemma}
\label{decomp}
Let $N$ be a connected group and denote by $X$ the (possibly empty) set of all non-trivial  continuous homomorphisms from $N$ to $\mathbb{T}$. Then there exists a subset $S \subset X$ such that
\[
X = S \cup \check{S} \quad \textrm{and} \quad S \cap \check{S} = \emptyset,
\]
where
\[
\check{S} = \Big\{ \xi \in X \, : \, \check{\xi} \in S \Big\}.
\]
Furthermore, if $N$ in addition is second countable, then for any
set $S$ as above, there exists a Borel measurable map
$\epsilon : \operatorname{Aut}(N) \rightarrow \{-1,1\}$ such that
\[
\epsilon(\alpha) \circ \xi \circ \alpha \in S
\]
for all $\alpha \in \operatorname{Aut}(N)$.
\end{lemma}

\begin{proof}
For the first assertion, it suffices to show that the self-map $\xi \mapsto \check{\xi}$ does not admit a fixed point in $X$. However, the equation $\check{\xi} = \xi$ holds if and only
if the image of $\xi$ is contained in the \emph{finite} subgroup of order two elements of $\mathbb{T}$, which is impossible since $N$ is connected.

For the second assertion, we note that since $N$ is second countable, the
Hilbert space $L^2(N,m_N)$ is separable and since every two distinct
element in $X$ are orthogonal in $L^2(N,m_N)$, we see that $X$ (and
thus $S$) is at most countable. In particular, the set
\[
E = \Big\{ \alpha \in \operatorname{Aut}(N) \, : \, \xi \circ \alpha \in S \Big\}
\]
is $F_\sigma$ (so in particular Borel measurable), and thus the map defined by
$\epsilon(\alpha) = 1$ if $\alpha \in E$ and $\epsilon(\alpha) = -1$
if $\alpha \notin E$ is Borel measurable and satisfies
$\epsilon(\alpha) \circ \xi \circ \alpha \in S$ for all $\alpha$ in $\operatorname{Aut}(N)$.
\end{proof}

The third and final lemma before we embark on the proof of Proposition \ref{prop1} is technical but quite important. It will be used to show that
certain maps which we construct, without any a priori regularity whatsoever, must in fact be Borel measurable.

\begin{lemma}
\label{borel}
Let $G$ and $M$ be compact groups and let $N$ be a closed
subgroup of $G$. Let $C \subset G$ and $I \subset M$ be Borel sets and
suppose that there exist a continuous surjective homomorphism
$\xi : N \rightarrow M$, a conull Borel set $X \subset G$ and a map
$\gamma : X \rightarrow M$ such that
\[
x^{-1}C \cap N \sim \xi^{-1}(I\gamma(x))
\]
for all $x \in X$. If $M$ is second countable and the essential stabilzer group of $I \subset M$ is trivial, then $\gamma$ is Borel measurable
on $X$.
\end{lemma}

\begin{proof}
Fix a countable basis $(U_n)$ for the topology on $M$ and note that
\[
x^{-1} C \cap N \cap \xi^{-1}(U_n) \sim \xi^{-1}(I \gamma(x) \cap U_n)
\]
for all $n$ and for all $x$ in $X$. Define the maps
\[
\Psi : M \rightarrow [0,1]^{\mathbb{N}} \quad \textrm{and} \quad \Phi : M \rightarrow [0,1]^{\mathbb{N}}
\]
by
\[
\Psi(t)_n = m_M(I \, t \cap U_n)
\quad \textrm{and} \quad
\Phi(x)_n = m_N\big(x^{-1}C \cap N \cap \xi^{-1}(U_n)\big)
\]
for $t \in M$ and $x \in G$ and $n$ in $\mathbb{N}$.

Clearly, both $\Psi$ and $\Phi$ are
Borel measurable with respect to the product Borel structure on
$[0,1]^{\mathbb{N}}$ and by construction, we have
\[
\Psi(\gamma(x)) = \Phi(x), \quad \forall \, x \in G.
\]
Since $C$ has trivial essential stabilizer, Corollary \ref{ess. stab} in
Appendix I of this paper asserts that $\Psi$ is injective, and is thus a Borel isomorphism onto its image (see e.g. Theorem A.4 in \cite{Zi}). We
conclude that the composition $\gamma = \Psi^{-1} \circ \Phi$ is
Borel measurable on $X$.
\end{proof}

\subsection{Proof of Proposition \ref{prop1}}
Let $G$ be a compact and second countable group with a split
identity component $N$, i.e. the group $G$ can be written as a
semi-direct product
\[
G = N \rtimes Q,
\]
where $Q < \operatorname{Aut}(N)$ is a closed subgroup isomorphic to the quotient group $G/N$. In particular,
\[
m_G(C)
=
\int_Q m_{N}\Big(p^{-1} \, C \cap N\Big) \, dm_Q(p)
=
\int_Q m_{N}\Big(C \, q^{-1} \cap N\Big) \, dm_Q(q)
\]
for every Borel set $C \subset G$, where $m_N$ and $m_Q$ denote the
Haar probability measures on $N$ and $Q$ respectively.

Suppose that $(A,B)$ is a critical pair in $G$ with respect to $N$. Recall
that this means that there exists a conull subset $Z \subset Q \rtimes Q$
such that
\[
\big(m_G\big)_{pq}^N(AB)
=
\big(m_G\big)_{p}^N(A) + \big(m_G\big)_{q}^N(B) < 1
\]
and
\[
\big(m_G\big)_{p}^N(A) = m_G(A)
\quad \textrm{and} \quad
\big(m_G\big)_{q}^N(B) = m_G(B)
\]
for all $(p,q)$ in $Z$. In our setting, these identities imply that
\[
m_N\Big( \big(p^{-1}A \cap N\big) \, \big(Bq^{-1} \cap N\big)\Big)
\leq
\big(m_G\big)_{pq}^N\Big(AB\Big)
=
m_N\Big( p^{-1} \, AB \, q^{-1} \cap N\Big)
\]
and
\[
\big(m_G\big)_{p}^N(A) + \big(m_G\big)_{q}^N(B)
=
m_N(p^{-1} \, A \cap N) + m_N(B \, q^{-1} \cap N),
\]
and thus
\[
\big(p^{-1}A \cap N\big) \, \big(Bq^{-1} \cap N\big)
\sim
p^{-1} \, AB \, q^{-1} \cap N
\]
and
\[
m_N\Big( \big(p^{-1}A \cap N\big) \, \big(Bq^{-1} \cap N\big)\Big)
=
m_N(p^{-1} \, A \cap N) + m_N(B \, q^{-1} \cap N)
\]
since $N$ is connected, and thus cannot admit sub-critical pairs by
Kneser's Theorem.

In particular, we see that
\[
\Big(p^{-1}A \cap N, B \, q^{-1} \cap N \Big)
\]
is a critical pair in $N$ and
\[
m_N\Big(p^{-1}A \cap N\Big) = m_G(A)
\quad \textrm{and} \quad
m_N\Big(B \, q^{-1} \cap N\Big) = m_G(B)
\]
for all $(p,q) \in Z$. Let $I_o$ and $J_o$ denote the (uniquely defined) closed and symmetric intervals in $\mathbb{T}$ with Haar measures equal to
$m_G(A)$ and $m_G(B)$ respectively, and let $\Sigma$ denote the set
of non-trivial homomorphisms from $N$ into $\mathbb{T}$.

Since $N$ is a Kneser group, there exist maps (with no a priori Borel regularity
whatsoever)
\[
\xi : Z \rightarrow \Sigma \quad \textrm{and} \quad \alpha, \beta : Z \rightarrow \mathbb{T}
\]
such that
\[
p^{-1} \, A \cap N \subset \xi_z^{-1}\Big(I_o \, \alpha(z) \Big)
\quad \textrm{and} \quad
B \, q^{-1} \cap N \subset \xi_z^{-1}\Big(J_o \, \beta(z) \Big)
\]
and
\[
p^{-1}ABq^{-1} \cap N \sim \xi_{z}^{-1}\Big(I_o \, J_o \, \alpha(z) \, \beta(z) \Big).
\]

\begin{step}[Producing Borel regularity]
By the first part of Lemma \ref{decomp} there exists a set $S \subset \Sigma$ such that
\[
\Sigma = S \sqcup \check{S}.
\]
Since
\[
\xi^{-1}(I) = \check{\xi}^{-1}(I^{-1}), \quad \forall \, \xi \in \Sigma
\]
and for every set $I \subset \mathbb{T}$, we may without
loss of generality (upon possible modifying $\alpha$ and $\beta$) assume that $\xi_z \in S$ for all $z \in Z$. \\

Note that
\[
\xi_z^{-1}\Big(I_o \, \alpha(z)\Big)
\sim
\xi_{z'}^{-1}\Big(I_o \, \alpha(z')\Big)
\]
whenever $z = (p,q)$ and $z' = (p,q')$ are contained $Z$ and
\[
\xi_z^{-1}\Big(J_o \, \beta(z)\Big)
\sim
\xi_{z'}^{-1}\Big(J_o \, \beta(z')\Big)
\]
whenever $z = (p,q)$ and $z' = (p',q)$ are contained in $Z$. \\

Since $\xi_{z}$ and $\xi_{z'}$ belong to $S$, Lemma \ref{sim}
guarantees that if $z = (p,q)$ and $z' = (p',q')$ belong to $Z$,
then
\[
\xi_z = \xi_{z'} \quad \textrm{and} \quad \alpha(z) = \alpha(z')
\]
if $p = p'$ and
\[
\xi_z = \xi_{z'} \quad \textrm{and} \quad \beta(z) = \beta(z')
\]
if $q = q'$, so we conclude that there exist $\xi$ in $S$,
conull Borel sets
\[
X \subset G/N \quad \textrm{and} \quad Y \subset X \times X
\]
and maps $\alpha_o, \beta_o : X \rightarrow \mathbb{T}$ such that
\[
\xi_z = \xi \quad \textrm{and} \quad \alpha(z) = \alpha_o(p) \quad \textrm{and} \quad \beta(z) = \beta_o(q)
\]
for all $z = (p,q)$ in $Y$. \\

Hence,
\[
p^{-1} \, A \cap N \subset \xi^{-1}\Big(I_o \, \alpha_o(p)\Big)
\quad \textrm{and} \quad
B \, q^{-1} \cap N \subset \xi^{-1}\Big(J_o \, \beta_o(p)\Big)
\]
and
\[
p^{-1}ABq^{-1} \cap N \sim \xi^{-1}\Big(I_o \, J_o \, \alpha_o(p) \, \beta_o(q)\Big)
\]
for all $(p,q) \in Y$. By Lemma \ref{borel}, we conclude that the maps $\alpha_o$ and $\beta_o$ are Borel measurable on $X$.
\end{step}

\begin{step}[Almost conjugacy invariance of $\xi$]
We can rewrite the last identities on the form
\[
A \cap Np
\subset
\Big(\xi \circ c_{p^{-1}}\Big)^{-1}\Big(I_o \, \alpha_o(p)\Big)p
\quad \textrm{and} \quad
B \cap Nq
\subset
\xi^{-1}\Big(J_o \, \beta_o(q)\Big)q
\]
and
\[
AB \cap Npq \sim \Big(\xi \circ c_{p^{-1}}\Big)^{-1}\Big(I_o \, J_o \, \alpha_o(p) \, \beta_o(q)\Big) \, pq,
\]
where $(p,q) \in Z$, and $c_p$ denotes the automorphism of $N$ induced
by $p$ in $Q$. \\

By the second part of Lemma \ref{decomp}, we can find a Borel measurable
map $\epsilon : Q \rightarrow \{-1,1\}$ such that
\[
\epsilon_p \circ \xi \circ c_{p^{-1}} \in S, \quad \forall \, p \in Q,
\]
where $\epsilon_p = \epsilon(c_{p^{-1}})$ and since $I_o$ and $J_o$ are assumed to be symmetric intervals, we have
\[
A \cap Np
\subset
\Big(\epsilon \circ \xi \circ c_{p^{-1}}\Big)^{-1}\Big(I_o \, \alpha_o(p)^{\epsilon_p}\Big)p
\quad \textrm{and} \quad
B \cap Nq
\subset
\xi^{-1}\Big(J_o \, \beta_o(q)\Big)q
\]
and
\[
AB \cap Npq \sim \Big(\epsilon_p \circ \xi \circ c_{p^{-1}}\Big)^{-1}\Big(I_o \, J_o \, \big(\alpha_o(p) \, \beta_o(q)\big)^{\epsilon_p}\Big) \, pq,
\]
for all $(p,q) \in Y$. Since
\[
\epsilon_p \circ \xi \circ c_{p^{-1}} \in S
\]
for almost all $p$, we conclude by Lemma \ref{sim} that
\[
\epsilon_p \circ \xi \circ c_{p^{-1}} = \epsilon_{p'} \circ \xi \circ c_{{p'}^{-1}}
\]
and
\[
\Big(\alpha_o(p) \, \beta_o(q)\Big)^{\epsilon_p}
=
\Big(\alpha_o(p') \, \beta_o(q')\Big)^{\epsilon_{p'}}
\]
for all $(p,q)$ and $(p',q')$ in $Y$ such that $pq = p'q'$. The first
of these equations clearly implies that $\xi \circ c_{p^{-1}}$ is
either $\xi$ or $\check{\xi}$ for almost every (and hence all) $p$ in
$Q$. In particular,
\[
\epsilon_p \circ \xi \circ c_{p^{-1}} = \xi, \quad \forall \, p \in Q,
\]
which readily implies that the map $p \mapsto \epsilon_p$ is a Borel measurable (hence continuous) homomorphism , and this finishes the proof of Proposition \ref{prop1}.
\end{step}

\section{Appendix I: Relativization of critical pairs}

The aim of this appendix is to establish the principle of relativization of critical
pairs formulated in the introduction of this paper.

Let $G$ be a compact and second countable group. Recall that the
\emph{identity component} $G^o$ of $G$ is defined as the
intersection of all open subgroups of $G$. Since $G$ is assumed to
be compact and second countable, we can find a decreasing
\emph{sequence} $(U_n)$ of open and \emph{normal} subgroups
such that
\[
G^o = \bigcap_{n} U_n.
\]
Clearly, $G^o$ is a closed (but not necessarily open) normal subgroup of $G$.

Let $m_G$ and $m_{G^o}$ denote the Haar probability measures on $G$ and $G^o$
and note that we may regard $m_{G^o}$ as a Borel probability measure on $G$
supported on the subgroup $G^o$. By uniqueness of Haar probability measures
on compact groups, we have
\[
m_G = \int_{G} (r_x)_*m_{G^o} \, dm_{G}(x),
\]
where $r_x : G \rightarrow G$ denotes the right multiplication on $G$ by
the element $x$ in $G$. We shall also write
\[
\big(m_G\big)_x^{G_o} = (r_x)_*m_{G^o}
\]
for $x$ in $G$.

Recall that a pair $(A,B)$ of Borel sets in $G$ is \emph{critical with respect to $G^o$}
if there exist conull Borel sets
\[
X \subset G \quad \textrm{and} \quad Z \subset X \times X
\]
such that
\[
m_G(A) = \big(m_G\big)_x^{G_o}(A)
\quad \textrm{and} \quad
m_G(B) = \big(m_G\big)_x^{G_o}(B)
\]
for all $x, y \in X$ and
\[
\big(m_G\big)_{xy}^{G_o}(AB)
=
\big(m_G\big)_{x}^{G_o}(A) + \big(m_G\big)_{y}^{G_o}(B)
\]
for all $(x,y)$ in $Z$. Note that if a pair $(A,B)$ is critical with respect to $G^o$,
then in particular it is critical. \\

More generally, if $N < G$ is any closed subgroup of $G$, we can define
\[
\big(m_G\big)_x^N = (r_x)_*m_{N}, \quad x \in G,
\]
and we still have
\[
m_G = \int_G (r_x)_*m_{N} \, dm_G(x).
\]
There is an obvious extension of the notion of a critical pair with respect to $N$
defined above. In the important special case when $N$ is an \emph{open and normal}
subgroup of $G$, this notion can be described as follows. There exist conull
Borel sets
\[
X \subset G \quad \textrm{and} \quad Z \subset G \times G
\]
such that
\[
m_G(A) = \frac{m_G(A \cap Ux)}{m_G(U)}
\quad \textrm{and} \quad
m_G(B) = \frac{m_G(B \cap Uy)}{m_G(U)}
\]
for all $x, y \in X$ and
\[
m_G(AB \cap xUy) = m_G(A \cap xU) + m_G(B \cap Uy)
\]
for all $(x,y) \in Z$. \\

Let $U$ be an open and normal subgroup of $G$. A pair $(A,B)$ of Borel sets in
$G$ is said to be \emph{almost sub-critical in $U$}, if there exist conull Borel
sets
\[
A' \subset A \quad \textrm{and} \quad B' \subset B
\]
and $p$ and $q$ in $G$ such that the pair
\[
\Big(p^{-1}A' \cap U,B'q^{-1} \cap U \Big)
\]
is sub-critical. If $(A,B)$ is almost sub-critical with respect to \emph{some}
open and normal subgroup of $G$, then we shall simply say that $(A,B)$ is
\emph{locally sub-critical}. \\

The main result of this appendix can now be formulated as follows.

\begin{proposition}[Relativization of critical pairs]
\label{relative}
Suppose $(A,B)$ is a critical pair in $G$ which is not locally sub-critical. Then there
exist an open subgroup $L < G$ and conull $\sigma$-compact sets
\[
A' \subset A \quad \textrm{and} \quad B' \subset B
\]
and $s, t \in G$ such that
\[
sA' \subset L \quad \textrm{and} \quad B't \subset L
\]
and $(sA',B't)$ is critical in $L$ with respect to the identity component of $L$.
\end{proposition}

The proof of this proposition will be an immediate consequence of the following
two propositions, which we now describe. \\

Given an open and normal subgroup $U < G$ and a Haar measurable set
$C \subset G$, we define
\[
C_x = C \cap Ux \quad \textrm{and} \quad C^+ = \Big\{ y \in G/U \, : \, C_y \neq \emptyset \Big\} \subset G/U,
\]
for $x$ in $G$. We shall say that a $C \subset G$ is \emph{$U$-balanced} if
\[
e \in C^+ \quad \textrm{and} \quad m_G(C_y) > 0, \quad \forall \, y \in C^+.
\]
Clearly, if $C$ is $U$-balanced, then so is every conull subset of $C$. Finally,
we say that a pair $(A,B)$ is $U$-balanced if both $A$ and $B$ are $U$-balanced
sets.

\begin{proposition}
\label{rel1}
Let $U$ be an open normal subgroup of $G$ and
suppose that $(A,B)$ is a $U$-balanced critical
pair in $G$, which is not almost sub-critical in
$U$. Then $(A,B)$ is critical with respect to $U$.
\end{proposition}

Let $(U_n)$ be a decreasing sequence of open and normal subgroups of $G$
whose intersection equals the identity component of $G$ and suppose
$(A,B)$ is a critical pair in $G$. For every
$n$, we define
\[
A_n = A \cap A_n^{+}U_n \quad \textrm{and} \quad B_n = B \cap B_n^{+}U_n
\]
and note that
\[
m_G(A_n) = m_G(A) \quad \textrm{and} \quad m_G(B_n) = m_G(B)
\]
for all $n$. Hence, the intersections
\[
A' = \bigcap_n A_n \quad \textrm{and} \quad B' = \bigcap_n B_n
\]
are conull subsets of $A$ and $B$ respectively, and by construction, they
satisfy
\[
m_G(A' \cap U_n x) > 0 \quad \textrm{and} \quad m_G(B' \cap U_n y) > 0
\]
for all $x \in A'^{+}$ and $y \in B'^{+}$. Upon passing to conull
$\sigma$-compact subsets and translating, we see that for every
pair $(A,B)$ of Haar measurable sets, there exist conull
$\sigma$-compact subsets $A' \subset A$ and $B' \subset B$
and $s, t \in G$ such that $sA'$ and $B't$ are $U$-balanced sets,
and
\[
m_G(A'B') \leq m_G(A') + m_G(B').
\]
If the inequality would be strict, then $(A',B')$ is sub-critical and
thus $(A,B)$ would be locally sub-critical. \\

We conclude that if $(A,B)$ is a critical pair in $G$ which is not
locally sub-critical, then there exist conull $\sigma$-compact
sets
\[
A' \subset A \quad \textrm{and} \quad B' \subset B
\]
and $s, t \in G$ such that $(sA',B't)$ is $U_n$-balanced for every $n$
and thus, by Proposition \ref{rel1}, critical with respect to $U_n$
for every $n$. Proposition \ref{relative} now follows immediately
from the following result.

\begin{proposition}
\label{rel2}
Let $G$ be a compact group and
suppose that $(N_k)$ is a decreasing sequence of closed
normal subgroups of $G$ with intersection $N$. If
$(A,B)$ is a pair of Borel sets in $K$ which is critical with
respect to $N_k$ for every $k$, then $(A,B)$ is critical
with respect to $N$.
\end{proposition}

\subsection{Proof of Proposition \ref{rel1}}
Recall that if $U < G$ is an open subgroup of $G$ and $C \subset G$ is a Borel set, we define
\[
C^{+} = \Big\{ x \in G/U \, : \, C \cap Ux \neq \emptyset \Big\} \subset G/U.
\]
Proposition \ref{rel2} is now contained in the following result.

\begin{proposition}
\label{helpmain1.1}
Let $U$ be an open normal subgroup of $G$ and suppose $(A,B)$ is a
$U$-balanced pair of Borel sets in $G$. Fix $x_o \in A^{+}$ such that
\[
m_G(A_{x_o}) = \max_{x \in A^{+}} m_G(A_x).
\]
If $(A,B)$ is not almost sub-critical in $U$ and there exists
$z_o \in (AB)^{+} \setminus x_o B^{+}$, then
\[
m_G(AB) \geq m_G(A_xB_y) + \frac{|B^{+}|}{|A^{+}|} \cdot m_G(A)
+ m_G(B) ,
\]
for all $x \in A^{+}$ and $y \in B^{+}$ such that $xy = z_o$.
Furthermore, if the pair $(A,B)$ in addition is critical, then
\[
A^+ = B^{+}
\]
and $A^+$ is a subgroup of $G/U$ and $(A,B)$ is a critical pair in
the compact group $L = A^+U$ with respect to closed normal
subgroup $U$ of $L$.
\end{proposition}

\begin{proof}
Since $(A,B)$ is not almost sub-critical in $U$, we have
\[
m_G(A_x B_y) \geq m_G(A_x) + m_G(B_y) \quad \forall \, (x,y) \in A^{+} \times B^{+},
\]
and since $U$ is normal, we have
\[
AB = \bigsqcup_{z \in (AB)^{+}} AB \cap Uz
=
\bigsqcup_{z \in (AB)^{+}}
\left( \bigcup_{xy = z} A_x B_y\right).
\]
Pick $x_o$ in $A^{+}$ which satisfies
\[
m_G(A_{x_o}) = \max_{x \in A^{+}} m_G(A_x)
\]
and note that $m_G(A) \leq |A^{+}| \cdot m_G(A_{x_o})$. In particular, if $(A,B)$ is critical and not locally sub-critical, then
\begin{eqnarray*}
 |A^{+}| \cdot m_G(A_{x_o}) + m_G(B)
 &\geq &
 m_G(A) + m_G(B) =
 m_G(AB) \\
 &\geq &
 m_G\left(\bigcup_{y \in B^{+}} A_{x_o} \, B_y\right) \\
 &= &
 \sum_{y \in B^{+}} m_G(A_{x_o} B_y) \\
 &\geq &
  \sum_{y \in B^{+}} m_G(A_{x_o}) + m_G(B_y) \\
 &\geq &
 |B^{+}| \cdot m_G(A_{x_o}) + m_G(B),
\end{eqnarray*}
from which we can conclude that $|A^{+}| \geq |B^{+}|$. Arguing
symmetrically with $y_o$ in $B^{+}$ which satisfies
\[
m_G(B_{y_o}) = \max_{y \in B^{+}} m_G(B_y),
\]
we conclude that $|A^{+}| = |B^{+}|$ if $(A,B)$ is critical and not locally
sub-critical. \\

Suppose there exists $z_o \in (AB)^{+} \setminus x_o B^{+}$. Then, for
all $x \in A^{+}$ and $y \in B^{+}$ with $xy = z_o$, we have
\begin{eqnarray*}
m_G(AB)
&=&
\sum_{z \in (AB)^{+}} m_G\Big(\bigcup_{xy = z} A_x B_y \Big) \\
&\geq &
m_G(A_x B_y) + \sum_{z \in x_oB^{+}} m_G(A_{x_o} B_{x_o^{-1}z}) \\
&\geq &
m_G(A_x B_y) +
\sum_{z \in x_oB^{+}} m_G(A_{x_o}) + m_G(B_{x_o^{-1}z})
\\
&\geq &
m_G(A_x B_y) +
|B^{+}| \cdot m_G(A_{x_o}) + m_G(B) \\
&\geq &
m_G(A_x B_y) +
\frac{|B^{+}|}{|A^{+}|}\cdot m_G(A) + m_G(B) \\
&>&
\frac{|B^{+}|}{|A^{+}|}\cdot m_G(A) + m_G(B),
\end{eqnarray*}
since $m_G(A_x)$ and $m_G(B_y)$ are both positive. In particular, if
$(A,B)$ is a critical, and thus $|A^{+}| = |B^{+}|$, we see that the above
chain of inequalities is impossible, so we must conclude that
\[
(AB)^{+} = A^{+} \, B^{+} = x_o \, B^{+}.
\]
Since $A$ and $B$ are $U$-balanced, and thus $e \in A^{+} \cap B^{+}$, we
have
\[
B^{+} =x_o^{-1}A^{+} B^{+} \supset x_o^{-1}A^{+}
\quad \textrm{and} \quad
x_o^{-1}A^+ = B^{+} \quad \textrm{and} \quad B^+ B^{+} = B^{+},
\]
since $|A^{+}| = |B^{+}|$. Put differently, $B^{+}$ is a semigroup of the \emph{finite} group $G/U$. Since any semigroup of a finite group is
in fact a subgroup, this shows that $A^{+} = B^{+}$ and thus, by a closer study of the series of inequalities above (and arguing symmetrically with
the roles of $A$ and $B$ and $x_o$ and $y_o$ interchanged), we conclude that
\[
m_G(A \cap Ux) = \frac{m_G(A)}{|A^{+}|}
\quad \textrm{and} \quad
m_G(B \cap Uy) = \frac{m_G(B)}{|B^{+}|}
\]
and
\[
m_G(AB \cap Uxy) = m_G(A \cap Ux) + m_G(B \cap Uy)
\]
for all $x, y \in L = A^{+}U$.  Hence $(A,B)$ is a critical pair in $L$
with respect to $U$ (note that this does not mean that $(A,B)$ is a
critical pair in $G$ with respect to $U$, unless of course $A^{+} = G/U$).
\end{proof}

\subsection{Proof of Proposition \ref{rel2}}
The aim of this subsection is to show that criticality with respect to a decreasing
sequence of closed normal subgroups is inherited by the intersection of these
subgroups. This will follow from rather standard differentiation lemmata which
we prove here for completeness. \\

Recall that if $G$ is a compact group and $x \in G$, we denote by $r_x : G \rightarrow G$
the right multiplication in $G$ by $x$. Clearly, this map induces a continuous
selfmap on the space of probability measures on $G$ equipped with the weak*-topology.

\begin{proposition}
\label{rel1.1}
Suppose $(\mu_n)$ is a sequence of Borel probability measures on $G$ with weak*-limit
$\mu$. For every Borel set
$B \subset G$ and subsequence $(n_k)$, there exist a further
subsequence $(n_{k_j})$ and a Haar conull set $X \subset G$
such that
\[
\lim_{j \rightarrow \infty}
\big(r_x\big)_*\mu_{n_{k_j}}(B)
=
\big(r_x\big)_*\mu(B)
\]
for all $x \in X$.
\end{proposition}

\begin{proof}
Define the functions
\[
\psi_{n}(x) = \big(r_x\big)_*\mu_{n}(B)
\quad \textrm{and} \quad
\psi(x) = \big(r_x\big)_*\mu(B), \quad \textrm{and} \quad x \in G.
\]
By Theorem 444F(e) in \cite{Fr}, these functions are Borel measurable on $G$, so
it suffices to show (and then pass to an almost everywhere convergent subsequence) that $\psi_n \rightarrow \psi$ in $L^2(G,m_G)$ (in the norm topology),
or equivalently, that $\psi_n \rightarrow \psi$ (in the weak topology)
\emph{and}
\[
\lim_{n \rightarrow \infty} \int_G \big| \psi_n(x) \big|^2 \, dm_G(x)
=
\int_G \big| \psi(x) \big|^2 \, dm_G(x).
\]
Fix $\varphi$ in $L^2(X,m_G)$ and note that
\[
\int_G \varphi(x) \, \psi_n(x) \, dm_G(x)
=
\int_G \Big( \int_G \varphi(x) \, \chi_B(yx^{-1}) \, dm_G(x) \Big) \, d\mu_n(y)
\]
by Fubini's Theorem, and the function
\[
\eta_\varphi(y) = \int_G \varphi(x) \, \chi_B(yx^{-1}) \, dm_G(x), \quad y \in G,
\]
is continuous on $G$. Since $\mu_n \rightarrow \mu$ in the weak*-topology, we
have
\[
\int_G \varphi(x) \, \psi_n(x) \, dm_G(x) = \int_G \eta_\varphi \, d\mu_n
\rightarrow
\int_G \eta_\varphi \, d\mu = \int_G \varphi(x) \, \psi(x) \, dm_G(x).
\]
Since $\varphi$ is arbitrary, we conclude that $\psi_n \rightarrow \psi$ weakly in $L^2(G,m_G)$.

The proof that the norms converge as well follows very similar lines as the
proof of the weak convergence and is omitted.
\end{proof}

\begin{corollary}
\label{conv subgroups}
Suppose $(N_k)$ is a decreasing sequence of closed subgroups of $G$ with intersection $N$. For every Borel set $B \subset G$ and subsequence $(n_k)$,
there exist a further subsequence $(n_{k_j})$ and a Haar conull subset
$X \subset G$ such that
\[
\lim_{j \rightarrow \infty} \big(m_G)_x^{N_{k_j}}(B)
=
\big(m_G\big)_x^N(B)
\]
for all $x$ in $G$.
\end{corollary}

\begin{proof}
Recall that if $M$ is a closed subgroup of $G$, then
\[
\big(m_G\big)_x^{M} = \big(r_x\big)_*m_{M}, \quad \forall \, x \in G,
\]
where $m_M$ is considered as a Borel probability measure on $G$.
Since $m_{N_k} \rightarrow m_{N}$ in the weak*-topology on $G$, the corollary
follows immediately from Proposition \ref{rel1.1}.
\end{proof}

Recall that if $B \subset G$ is a Borel set, then the
\emph{essential stabilizer} of $B$ is the closed subgroup
\[
E_B = \Big\{ g \in G \, : \, m_G(B \Delta g B) = 0 \Big\},
\]
where $\Delta$ denotes the symmetric set difference.

\begin{corollary}
\label{ess. stab}
Let $G$ be a compact and second countable group and suppose
$(U_n)$ is a countable basis for the topology of $G$. If $A, B \subset G$
are pre-compact Borel sets such that
\[
m_G(A \cap U_n) = m_G(B \cap U_n)
\]
for all $n$, then $A \sim B$. In particular, if $C \subset G$ is a
pre-compact Borel set and there exist $x, y \in G$ such that
\[
m_G(Cx \cap U_n) = m_G(Cy \cap U_n)
\]
for all $n$, then $xy^{-1}$ belongs to the essential stabilizer of the set $C$.
\end{corollary}

\begin{proof}
Let $(V_n)$ be a decreasing sequence of open neighborhoods of the identity
in $G$. The assumption above in particular implies that
\[
\frac{m_G(A \, x^{-1} \cap V_n)}{m_G(V_n)} = \frac{m_G(B \, x^{-1} \cap V_n)}{m_G(V_n)}, \quad \forall \, x \in G,
\]
for all $n$, or equivalently, that $(r_x)_*\mu_n(A) = (r_x)_*\mu_n(B)$ for all $n$ and
for all $x$ in $G$,
where
\[
\mu_n(\cdot) = \frac{m_G( \cdot \cap V_n)}{m_G(V_n)}.
\]
One readily shows that $\mu_n \rightarrow \delta_e$ in the weak*-topology on the space of
probability measures on $G$ and thus
$\chi_A(x) = \chi_B(x)$ for Haar almost every $x$ in $G$ by Proposition \ref{rel1.1}, which finishes the
proof.
\end{proof}

\subsubsection{Proof of Proposition \ref{rel2}}
Let $G$ be a compact group and let $(N_k)$ be a
decreasing sequence of closed normal subgroups
in $G$ with intersection $N$. Suppose $(A,B)$ is
a pair of Borel sets in $G$, which is critical with
respect to $N_k$ for every $k$, i.e. there exist
Borel sets
\[
X_k \subset G \quad \textrm{and} \quad Z_k \subset X_k \times X_k
\]
of full Haar measures such that
\[
m_G(A) = \big(m_G\big)^{N_k}_x(A)
\quad \textrm{and} \quad
m_G(B) = \big(m_G\big)^{N_k}_y(B)
\]
for all $x, y \in X_k$ and
\[
\big(m_G\big)^{N_k}_{xy}(AB) = \big(m_G\big)^{N_k}_x(A) + \big(m_G\big)^{N_k}_y(B)
\]
for all $(x,y)$ in $Z_k$. By Corollary \ref{conv subgroups}, there exist
a Haar conull set $X' \subset G$ and a subsequence $(k_j)$ such that
\[
\big(m_G\big)^{N}_x(A) = \lim_{j \rightarrow \infty} \big(m_G\big)^{N_{k_j}}_x(A)
\quad \textrm{and} \quad
\big(m_G\big)^{N}_y(B) = \lim_{j \rightarrow \infty} \big(m_G\big)^{N_{k_j}}_y(B)
\]
and
\[
\big(m_G\big)^{N}_z(AB) = \lim_{j \rightarrow \infty} \big(m_G\big)^{N_{k_j}}_z(AB)
\]
for all $x,y,z \in X'$. In particular,
\[
m_G(A) = \big(m_G\big)^{N}_x(A)
\quad \textrm{and} \quad
m_G(B) = \big(m_G\big)^{N}_y(B)
\]
and
\[
\big(m_G\big)^{N}_{xy}(AB) = \big(m_G\big)^{N}_x(A) + \big(m_G\big)^{N}_y(B)
\]
for all $(x,y)$ in $Z$, where
\[
X = X' \cap \Big( \bigcap_{k} X_k \Big)
\quad \textrm{and} \quad
Z = p^{-1}(X') \cap \Big( \bigcap_k Z_k \Big)
\]
and $p : G \times G \rightarrow G$ denotes the product map. This shows that $(A,B)$
is critical with respect to $N$.

\section{Appendix II: Rigidity of critical pairs}
The aim of this appendix is to establish the rigidity
of critical pairs
formulated in the introduction, namely the principle (which was
essentially pointed out already by Kneser in \cite{K56}) that
critical pairs which are sufficiently "regular" must be "maximal",
in the sense that any other critical pair which almost reduces to it,
must in fact reduce to it. \\

Let $G$ and $M$ be a compact groups. Recall that a closed set $I \subset M$
is \emph{regular} if it equals the closure of its interior, and we say that a
pair $(I,J)$ of Borel sets in $M$ is \emph{left stable} if the inclusion
$xJ \subset IJ$ implies that $x \in I$. There is an obvious analogous
notion of a right stable pair, and we say that $(I,J)$ is \emph{stable} if it is
both left and right stable. \\

Suppose $(I,J)$ is a stable pair in $M$ and there exists a compact subgroup
$K < \operatorname{Aut}(M)$ which fixes $I$ and $J$. Then it is not hard to show that
\[
\Big(I \rtimes K, J \rtimes K\Big)
\]
is stable in the group $M \rtimes K$, and if $\pi : G \rightarrow M \rtimes K$ is a continuous
surjective homomorphism, then
\[
\Big(p^{-1}\big(I \rtimes K\big), p^{-1}\big(J \rtimes K\big)\Big)
\]
is stable in $G$. Finally, since $\pi$ is necessarily open, we see that if $I$ and $J$
are regular sets, then so are the sets
\[
p^{-1}\big(I \rtimes K\big) \quad \textrm{and} \quad p^{-1}\big(J \rtimes K\big).
\]
In particular, the principle of rigidity of critical pairs mentioned in the introduction
follows immediately from the following result.

\begin{proposition}
\label{main4}
Suppose $(A_1,B_1)$ and $(A_2,B_2)$ are critical pairs in $G$ such that
\[
A_1 \sim A_2 \quad \textrm{and} \quad B_1 \sim B_2.
\]
If $(A_2,B_2)$ is stable and $A_2$ and $B_2$ are regular sets, then
\[
A_1 \subset A_2 \quad \textrm{and} \quad B_1 \subset B_2.
\]
\end{proposition}

\begin{proof}
Define the sets
\[
A_o = A_1 \cap A_2 \quad \textrm{and} \quad B_o = B_1 \cap B_2
\]
and note that
\[
m_G(A_o) = m_G(A_1) = m_G(A_2)
\quad \textrm{and} \quad
m_G(B_o) = m_G(B_1) = m_G(B_2),
\]
and
\begin{eqnarray*}
m_G(A_oB_o)
&\leq &
m_G(A_1B_1) \nonumber \\
&= &
m_G(A_1) + m_G(B_1) \nonumber \\
&= &
m_G(A_o) + m_G(B_o).
\end{eqnarray*}
If the first inequality would be strict, then $(A_o,B_o)$ is sub-critical,
and thus, by Kemperman's Theorem, there exists an open subgroup
of $G$ such that $A_o B_o = A_o U B_o$. Define
\[
V_1 = A_2^o \setminus A_o U \quad \textrm{and} \quad V_2 = B_2^o \setminus U B_o.
\]
Since $A_o U$ and $U B_o$ are clopen in $G$ and
\[
m_G(A_2^o) = m_G(A_2) \quad \textrm{and} \quad m_G(B_2^o) = m_G(B_2),
\]
we see that $V_1$ and $V_2$ are open null sets sets and thus empty. Since
$A_2$ and $B_2$ are regular sets, we conclude that
\[
A_2 = \overline{A_2^{o}} \subset A_o U \quad \textrm{and} \quad B_2 = \overline{B_2^{o}} \subset U B_o,
\]
and
\[
A_2 B_2 \subset A_o U B_o = A_o B_o \subset A_2 B_2.
\]
In particular, we see that $(A_o,B_o)$ cannot be sub-critical, and so
we can henceforth assume that
\[
m_G(A_oB_o) = m_G(A_1B_1) = m_G(A_2B_2).
\]
We wish to prove that $A_1 \subset A_2$ and $B_1 \subset B_2$.
Assume for the sake of contradiction that there exists
an element $x \in A_1 \setminus A_2$. Then,
\begin{eqnarray*}
m_G(A_1) + m_G(B_1)
&= &
m_G(A_1B_1) \\
&\geq &
m_G((A_o \cup \{x\})B_o) \\
&= &
m_G((A_oB_o \cup (xB_o \setminus A_oB_o)) \\
&= &
m_G(A_o B_o) + m_G(xB_o \setminus A_oB_o)) \\
&= &
m_G(A_o) + m_G(B_o) + m_G(xB_2 \setminus A_2B_2)) \\
&= &
m_G(A_1) + m_G(B_1) + m_G(xB_2 \setminus A_2B_2),
\end{eqnarray*}
which forces the set $xB_2 \setminus A_2 B_2$ to be a null set.
Since $A_2$ and $B_2$ are closed sets, so is $A_2 B_2$.
In particular, the set $x B_2^o \setminus A_2 B_2$ is open
and null, which forces it to be empty. Since $B_2$ is regular,
we have
\[
x\overline{B_2^o} = xB_2 \subset A_2 B_2.
\]
By assumption, $(A_2,B_2)$ is left stable, and thus $x \in A_2$,
which is a contradiction. This shows that $A_1 \subset A_2$. The proof
that $B_1 \subset B_2$ is completely analogous and we omit it.
\end{proof}

\section{Appendix III: Motivation and applications}

In this appendix we outline the motivation for the investigations
undertaken in this paper and describe some applications of the
main result. A significant part of the material in this section has
been developed in collaboration with A. Fish. \\

A classical and very useful fact, which is often attributed to Steinhaus,
asserts that the product of any two Borel measurable sets in a (locally) compact group with positive Haar measures has non-empty interior.
Many early results on small product sets in compact groups can be
viewed as attempts to quantify Steinhaus observation. For instance,
Kneser's Theorem can be restated along these lines as follows.
Let $G$ be a compact and connected abelian group and suppose $B, C \subset G$ are Borel
measurable subsets with positive Haar measures such that
\[
m_G(CB^{-1}) = m_G(C) + m_G(B) < 1.
\]
Then the product set $CB^{-1}$ contains a conull subset which is the pull-back of an open interval in $\mathbb{T}$ under a continuous surjective homomorphism $\xi : G \rightarrow \mathbb{T}$. Indeed, let $A = (CB^{-1})^c$
and note that
\[
m_G(AB) = m_G(A) + m_G(B) < 1.
\]
By Kneser's Theorem, there exists a continuous
surjective homomorphism $\xi : G \rightarrow \mathbb{T}$ and closed intervals $I$
and $J$ of $\mathbb{T}$ such that $A$ and $B$ are conull subsets of the
pull-backs $\xi^{-1}(I)$ and $\xi^{-1}(J)$ respectively, and thus
\[
A^c = CB^{-1} \supset \xi^{-1}(I^c).
\]
After the apperance of Steinhaus result in \cite{St}, many attempts were made to establish natural analogs of the phenomenon in discrete groups. One of the earliest succesful attempts in this direction was made by F\o lner in \cite{Fo}. To explain his result, we need to introduce some notation.

Recall that a countable group is \emph{amenable} if there exists a left
$G$-invariant positive and normalized functional $\lambda$ on
$\ell(G)$. Note that $\lambda$ gives rise to a left $G$-invariant
finitely additive probability measure $\lambda'$ on $G$ by
\[
\lambda'(C) = \lambda(\chi_C), \quad C \subset G.
\]
Let $\mathcal{L}_G$ denote the set of all left invariant means on $G$ and note
that $\mathcal{L}_G$ is a weak*-compact convex subset of $\ell^{\infty}(G)^*$.
We stress that $\mathcal{L}_G$ is always a very large set if $G$ is infinite; the cardinality of its set of extremal points is equal to the power set of the continuum. If $\mathcal{C} \subset \mathcal{L}_G$, then we define the
\emph{upper} and \emph{lower $\mathcal{C}$-density} of a subset $C \subset G $
by
\[
d_{\mathcal{C}}^*(C) = \sup_{\lambda \in \mathcal{C}} \lambda'(C)
\quad \textrm{and} \quad
d^{\mathcal{C}}_*(C) = \inf_{\lambda \in \mathcal{C}} \lambda'(C),
\]
respectively. In particular, if $\mathcal{C} = \mathcal{L}_G$, then we shall simply
write
\[
d^* = d_{\mathcal{L}_G}^* \quad \textrm{and} \quad d_* = d_*^{\mathcal{L}_G}.
\]
In \cite{Fo}, F\o lner proves that if $B$ and $C$ are subsets of $G$
with $d^*(B) > 0$ and $d_*(C) > 0$, then there exists a compact group
$K$, a homomorphism $\tau : G \rightarrow K$ with dense image, an open set
$U \subset K$ and a set $T \subset G$ with $d_*(T) = 1$ such that
\[
CB^{-1} \supset \tau^{-1}(U) \cap T.
\]
One of the main open problems in this area of research is whether the
set $T$ above can be dispensed with, that is to say, whether the
containment of the \emph{Bohr set} $\tau^{-1}(U)$ in $CB^{-1}$ is \emph{global}. \\

Around the same time as F\o lner's Theorem came out, Kneser
\cite{K56} proved
that if $CB^{-1}$ is sufficiently small, then such a global containment
is indeed possible to deduce. More precisely, Kneser considered the set
$\mathcal{B} \subset \mathcal{L}_{\mathbb{Z}}$ of \emph{Birkhoff means} on the additive group
$\mathbb{Z}$ of integers, which consists of all accumulation points of the
sequence
\[
\lambda_n(\varphi) = \frac{1}{n} \sum_{k=1}^n \varphi(k), \quad \varphi \in \ell^{\infty}(\mathbb{Z})
\]
in the weak*-topology on $\ell^{\infty}(\mathbb{Z})$ (one readily checks that these
means are always invariant) and established the following
result.

\begin{knesers}
Suppose $A, B \subset \mathbb{N}$ satisfy
\[
d_*^{\mathcal{B}}(AB) < d_*^{\mathcal{B}}(A) + d_*^{\mathcal{B}}(B).
\]
Then $AB$ is a \emph{cofinite} subset of periodic set in $\mathbb{N}$.
\end{knesers}
In particular, if we set $C = (AB)^c$, then $C \subset \mathbb{N}$ and
\[
d_{\mathcal{B}}^*(CB^{-1}) < \min\big(1,d_{\mathcal{B}}^*(C) + d_*^{\mathcal{B}}(B)\big),
\]
and conversely, if this inequality holds, then
\[
d_*^{\mathcal{B}}(AB) < d_*^{\mathcal{B}}(A) + d_*^{\mathcal{B}}(B)
\]
with $A = (CB^{-1})^c$. We conclude that if $(C,B)$ satisfies the bound
above, then $CB^{-1}$ contains a periodic subset of the integers, that
is to say, a subset of the form $\tau^{-1}(U)$, where
$\tau : \mathbb{Z} \rightarrow \mathbb{Z}/n \, \mathbb{Z}$ is the canonical map and
$U \subset \mathbb{Z}/n \, \mathbb{Z}$ is non-empty. In particular, $CB^{-1}$
globally contains a Bohr set. \\

In a recent joint paper \cite{BF6} with A. Fish, the author develops a general
technique to establish global Bohr containment results for
products of two subsets $C'$ and $B'$ of a countable amenable group
$G$ which satisfy the much weaker bound
\[
d_*(C'B'^{-1}) \leq \min\big(1,d_*(C') + d^*(B')\big).
\]
Note that if we set $A' = (C'B'^{-1})^c$, then
\[
d_*(A'^{-1}C') \leq \min\big(1,d^*(A') + d_*(C')\big).
\]
In order to explain the relevance of Theorem \ref{thm1} and Corollary
\ref{cor1} to our approach, we shall formulate a weak version of one
of the technical innovations in our paper, which we refer to as a
\emph{correspondence principle for product sets}.

The idea is that one can associate, to every pair of subsets $(A',C')$
of a countable amenable group $G$, a compact and second countable
group $K$, a continuous homomorphism $\tau : G \rightarrow K$ with dense
image and Borel sets $\tilde{A}, \tilde{C}$ such that $A$ and $C$ are
\emph{essentially contained} in pull-backs of the sets $\tilde{A}$ and
$\tilde{C}$ under the map $\tau$. The exact formulation is quite
technical and will be given below.

However, before we state our Correspondence Principle, we wish to
point out an important feature of countable amenable groups, which
should explain our choice to concentrate on products in compact
groups which satisfy the assumptions of Corollary \ref{cor1}.

\begin{proposition}
Let $G$ be a compact group which contains a dense countable
amenable subgroup. Then the identity component of $G$ is
abelian.
\end{proposition}

\begin{proof}
The proof combines a series of classical observations. A good source
to the material used in the proof is the book \cite{HS}.

Firstly, recall that by Peter-Weyl's Theorem, there exist a decreasing
net $(N_\alpha)$ of closed subgroups in $G$ with trivial intersection
such that $G/N_\alpha$ is a Lie group for every $\alpha$, i.e. it
sits a closed subgroup inside a unitary group $U(n)$ for some $n$.
In particular, the identity component $G_\alpha$ of the quotient group
$G/N_\alpha$ has finite index in the $G/N_\alpha$.

Now, if $\Gamma$ is a dense countable amenable subgroup of $G$,
then, for every $\alpha$, there exists a finite index subgroup
$\Gamma_\alpha$ of $\Gamma$ which is projects onto a dense
countable amenable subgroup of $G_\alpha$. Let $R_\alpha$ denote
the solvable radical of $G_\alpha$. Then $S_\alpha = G_\alpha/R_\alpha$
is a connected semisimple Lie group which contains a dense countable
amenable subgroup, namely the projection of $\Gamma_\alpha$ onto
its image in $S_\alpha$.

If $S_\alpha$ is non-trivial, then, by Tits' alternative, the last dense
subgroup must contain a free group on two generators which would
contradict its amenability. We conclude that $G_\alpha$ equals its
solvable radical and is thus a compact solvable connected Lie group,
and hence abelian (by the structure theory of such groups).

We have shown that $G_\alpha$ is abelian for every $\alpha$. If we
denote by $V_\alpha$ the pull-back of $G_\alpha$ in $G$ under
the canonical map $p_\alpha$ from $G$ onto $G/N_\alpha$, then
we see that $V_\alpha$ is an open and closed subgroup of $G$ for
every $\alpha$, and thus the intersection $C$ of the family
$(V_\alpha)$ contains the identity component of $G$. Fix any two
elements $x$ and $y$ in $C$ and note that $p_\alpha$ kills the
commutator of $x$ and $y$ for every $\alpha$. Since the kernel
of $p_\alpha$ is equal to $N_\alpha$ and the intersection of all
the family $(N_\alpha)$ is trivial, we conclude that the commutator
of $x$ and $y$ is trivial in $G$, i.e. the closed subgroup $C$ is
abelian, which forces the identity component of $G$ to be abelian.
\end{proof}

We can now formulate our correspondence principle for product
sets. Let $G$ be a countable group and suppose $X$ is a compact
metrizable $G$-space, i.e. $X$ is a compact and metrizable space
equipped with an action of $G$ by homeomorphisms. We say that
a point $x_o$ in $X$ is \emph{$G$-transitive} if its orbit
$G \cdot x_o$ is dense in $X$. Given any subset $B \subset X$
and $x$ in $X$, we define the set $B_x \subset G$ by
\[
B_x = \Big\{ g \in G \, : \, g \cdot x \in B \Big\}.
\]
If $K$ is a compact group and $\tau : G \rightarrow K$ is a homomorphism
with dense image, then $K$ is a natural $G$-space, where $G$ acts
on $K$ by right translations via $\tau$. Note that every point in $K$
is $G$-transitive.

We may also consider the diagonal action of $G$ on the product
$X \times K$ and if $\mu$ is a $G$-invariant probability measure
on $X$, then we say a $G$-invariant probability $\eta$ on $X \times K$
is a \emph{joining} of $\mu$ and $m_K$ if the natural projections of
$\eta$ onto $X$ and $K$ equal $\mu$ and $m_K$ respectively. Finally,
recall that a $G$-invariant probability measure $\mu$
on a compact $G$-space is \emph{ergodic} if every $G$-invariant Borel set is either null or conull with respect to $\mu$.

\begin{correspondence}
Let $G$ be a countable amenable group and suppose $A', C' \subset G$.
Then there exist compact metrizable $G$-spaces $X$ and $Y$ with
$G$-transitive points $x_o$ and $y_o$ respectively, clopen
sets $A \subset X$ and $C \subset Y$ such that
\[
A' = A_{x_o} \quad \textrm{and} \quad C' = C_{y_o}.
\]
Furthermore, there exists ergodic $G$-invariant Borel probability
measures $\mu$ and $\nu$ on $X$ and $Y$ respectively such that
\[
d^*(A') = \mu(A) \quad \textrm{and} \quad d_*(C') \leq \nu(C).
\]
Moreover there exist a compact second countable group $K$ with an abelian
identity component, a homomorphism $\tau : G \rightarrow K$ with dense image, Borel measurable sets $\tilde{A}, \tilde{C} \subset K$
with
\[
m_K(\tilde{A}) \geq \nu(A) \quad \textrm{and} \quad m_K(\tilde{C}) \geq \nu(C)
\]
and ergodic $G$-invariant joinings $\eta$ on $X \times K$ of $\nu$ and $m_K$ and $\xi$ on $Y \times K$ of $\nu$ and $m_K$ respectively such
that
\[
A_x \subset \tau^{-1}(\tilde{A}\,s)
\quad \textrm{and} \quad
C_y \subset \tau^{-1}(\tilde{C}\,t)
\]
for almost every $(x,s)$ and $(y,t)$ with respect to $\eta$ and $\xi$
respectively and
\[
d_*(A'^{-1}C') \geq m_K(\tilde{A}^{-1} \tilde{C}).
\]
\end{correspondence}
In particular, if $G$ is a countable amenable group and $(A',C')$ is a
pair of subsets of $G$ which satisfy
\[
d_*(A'^{-1}C') \leq \min\big(1,d^*(A') + d_*(C')\big),
\]
then the pair $(\tilde{A},\tilde{C})$ above must satisfy
\[
m_K(\tilde{A}^{-1} \tilde{C}) \leq \min(1,m_K(\tilde{A}) + m_K(\tilde{C})),
\]
i.e. the pair $(\tilde{A}^{-1},\tilde{C})$ is either sub-critical or critical in
$K$. In the first case, we can use Kemperman's Theorem to show that
$A$ must be "locally" contained in a subset of $G$ which is invariant
under a finite index subgroup and in the second case, Corollary \ref{cor1}
tells us that $A$ either has large "chunks" which are invariant under a finite index subgroup or it must "locally" be contined in a sturmian set.

In either case, these results are \emph{local} in nature, and need to be
complemented with further analysis in order to establish global
containment. We refer the interested reader to the paper \cite{BF6} for
more details.


\section{Introduction}

Throughout this paper, all graphs are finite and simple. A vertex-deleted subgraph of a graph $X$ is a subgraph formed by deleting exactly one vertex  and all of its incident edges from $X$. The multiset of all vertex-deleted subgraphs of $X$ is called deck of $X$ and denoted by $D(X)$. Two graphs are called hypomorphic when they have the same deck. We use the notation $X \dot{\cong} Y$ to show that two graphs $X$ and $Y$ are hypomorphic. Moreover,  we denote a path of length $l$ (number of its edges) by $P_l$ and use the name $l$-path for such a path. For other concepts and notations not presented here, we refer to \cite{graph theory,distance}.

Kelly in 1957 \cite{Kellytree} showed that two hypomorphic trees are isomorphic. After him,  Ulam in 1960 asked the following question \cite{Ulam}:
\begin{quote}
``Suppose $A$ and $B$ are sets with $n$ elements for each ($n \geq 3$). A metric $\rho$ is given on $A$ with the property that $\rho(x,y)$ is either $1$ or $2$ whenever $x$ and $y$ are in $A$ and $x \neq y$. A similar metric is given on $B$. Now suppose that the $n - 1$ element subsets of $A$ and $B$ can be labeled, $A_1$, $\cdots$, $A_n$ and $B_1$, $\cdots$, $B_n$ in such a way that each $A_i$ is isometric to $B_i$. Does this force $A$ to be isometric to $B$?"
\end{quote}
In fact, Ulam asked that are two hypomorphic graphs isomorphic? Nash$-$Williams provides a counterexample to show that this conjecture is not true in infinite graphs and he discusses about kinds of recognizable infinite graphs (\cite{infinite}). Further, one has been found a lot of counterexamples for directed graphs, too. See \cite{directed} for details.

Throughout this paper, we assume that $n \geq 3$ and $G$, $H$ are two labeled graphs with vertex sets $\{v_1 , \cdots , v_n\}$ and $\{u_1 , \cdots , u_n\}$,  respectively, in such a way that for each $i$, $1 \leq i \leq n$, $G \setminus \{v_i\} \cong H \setminus \{u_i\}$. For simplicity, we assume that $G_i = G \setminus \{v_i\}$ and $H_i = H \setminus \{u_i\}$. Wall in her master dissertation \cite{Wall}, proved that if $G \dot{\cong} H$, then $|E(G)| = |E(H)|$ and their degree sequences are the same. Furthermore, she concluded that there is an isomorphism between two regular hypomorphic graphs. For the sake of completeness, we first give a  simple similar proof for some of the mentioned results.

\vskip 3mm

\begin{lem}\label{E}
For each $i$, $1 \leq i \leq n$, $|E(G)| = |E(H)|$ and $deg(v_i) = deg(u_i)$.
\end{lem}
\begin{proof}
Since for each $i$, $G_i \cong H_i$, $|E(G_i)| = |E(H_i)|$. In the following we denote this number with $E_i$. It is clear to see that
\begin{eqnarray}\label{EE}
|E(G)| = E_i + deg(v_i) \ \ &,& \ \ |E(H)| = E_i + deg(u_i).
\end{eqnarray}
Applying sigma on both sides of equalities in Equation (\ref{EE}) yields that $\sum_{i = 1}^n|E(G)|$ = $\sum_{i = 1}^nE_i$ + $\sum_{i = 1}^ndeg(v_i)$ and $\sum_{i = 1}^n|E(H)|$ = $\sum_{i = 1}^nE_i$ + $\sum_{i = 1}^ndeg(u_i)$. This proves that $n|E(G)|$ = $\sum_{i = 1}^nE_i$ + $2|E(G)|$ and $n|E(H)|$ = $\sum_{i = 1}^nE_i$ + $2|E(H)|$.  Therefore, $\sum_{i = 1}^nE_i$ = $(n - 2)|E(G)|$ and $\sum_{i = 1}^nE_i$ = $(n - 2)|E(H)|$.
The last relation shows that $|E(G)| = |E(H)|$ and so (1) implies that $deg(v_i) = deg(u_i)$.
\end{proof}

\vskip 3mm

\begin{cor}\label{degree sequences}
Assume that $G \dot{\cong} H$. Then
\begin{enumerate}
\item[a)] $G$ and $H$ have the same degree sequences.
\item[b)] $G$ is an Eulerian graph if and only if $H$ is Eulerian.
\end{enumerate}
\end{cor}
\begin{proof}
To prove (a), we apply Lemma \ref{E} and for part (b), we use the part (a)  and the fact that all vertices of an Eulerian graph have even degree \cite[Theorem 6.2.2.]{graph theory}.
\end{proof}

\vskip 3mm

In the next section, it is proved that the number of $l-$paths having $v_i$ and $u_i$ are equal. Also, one can see that the number of $l-$paths contains $v_i,v_j$ and $u_i,u_j$ are the same. Another conclusion is that two hypomorphic graphs are isomorphic.

\vskip 3mm

\section{Main Results}

\vskip 3mm

If $x$ and $y$ are two different vertices of an arbitrary graph $X$, then the number of paths of length $l$ in $X$ containing $x$ and both $x, y$ are denoted by $p_{X}(x , l)$ and $p_{X}(x , y, l)$, respectively.

\begin{lem}\label{implem}
Suppose $v_i$ and $v_j$ are distinct vertices in a graph $G$ and $1 \leq l \leq n - 2$. Then, $\sum_{i = 1 , i \neq j}^np_G(v_j , v_i , l)$ = $lp_G(v_j, l).$
\end{lem}
\begin{proof}
Define $A_j$ to be the  set  of  all  paths  of  $G$ of length  $l$  containing  $v_j$ and $A_{ji}$ to be the  set  of  all  paths of length  $l$  containing $v_j$ and $v_i$. Moreover,
\begin{eqnarray*}
B_j &:=& \{(a_j , \{v_{i_1} , \cdots , v_{i_l}\}) \ | \ a_j \in A_j \ , \ V(a_j) = \{v_j , v_{i_1} , \cdots , v_{i_l}\}\},\\
B_{ji} &:=& A_{ji} \times \{i\}.
\end{eqnarray*}

It is easy to see that if $i$, $j$ and $k$ are three distinct integers that $1 \leq i , j , k \leq n$, then $B_{ji} \cap B_{jk} = \emptyset$. This concludes that

\begin{eqnarray*}
\left|\bigcup_{i = 1 \ \& \ i \neq j}^nB_{ji}\right| &=& \sum_{i = 1 \ \& \ i \neq j}^n|B_{ji}|\\
&=& \sum_{i = 1 \ \& \ i \neq j}^n|A_{ji} \times \{i\}|\\
&=& \sum_{i = 1 \ \& \ i \neq j}^n|A_{ji}|\\
&=& \sum_{i = 1 \ \& \ i \neq j}^np_G(v_j , v_i , l).
\end{eqnarray*}
Moreover, one can easily observe that $|B_j| = |A_j| = p_G(v_j, l)$.
Each member of $B_j$ such as $(a_j , \{v_{i_1} , \cdots , v_{i_l}\})$ is corresponding one-to-one to an element of $B_{ji_r}$, $1 \leq r \leq l$. So, $\bigcup_{i = 1,  i \neq j}^nB_{ji}$ has $l$ copies of every member of $B_j$. Therefore, $l|B_j|$ = $|\bigcup_{i = 1, i \neq j}^nB_{ji}|$. This implies that $p_G(v_j , l)$ = $\frac{\sum_{i = 1, i \neq j}^np_G(v_j , v_i , l)}{l}$, which proves the Lemma.
\end{proof}

\vskip 3mm

For a graph $X$, a cutnode of $X$ is a vertex whose removal increases the number of components. Moreover,  a nonseparable graph is connected, without any cutnode and not to be a single point. A block of $X$ is a maximal nonseparable subgraph. For an arbitrary cutnode $x \in V(X)$, we mean $bl(x)$ by the number of blocks connected to $x$. If $x$ is not a cutnode, then we define $bl(x) = 1$.

\vskip 3mm

\begin{lem}\label{cutnodes}
Let $G$ and $H$ be two connected hypomorphic graphs. Then
\begin{enumerate}
\item[a)] $v_i$ is a cutnode if and only if $u_i$ is a cutnode.
\item[b)] $bl(v_i) = bl(u_i)$.
\end{enumerate}
\end{lem}

\begin{proof}

(a) \ Suppose that $v_i$ is a cutnode of $G$. So, $G_i$ is disconnected. If $u_i$ is not a cutnode of $H$, then $H_i$ is connected which is impossible. The converse  is similar. (b) \ Since $bl(v_i)$ is exactly the number of connected components of $G_i$ and  $G_i \cong H_i$, $bl(v_i) = bl(u_i)$, proving the lemma.
\end{proof}

\vskip 3mm

Harary in \cite{connected} demonstrated that connected  is reconstructible. For a graph $X$, the number of connected components of $X$ is denoted by $c(X)$. It is trivial to see that if $X$ is connected, then $c(X) = 1$. Now, we are ready to prove that the number of connected components of two hypomorphic graphs are the same.

\vskip 3mm

\begin{thm}
If $G \dot{\cong} H$, then $c(G) = c(H)$.
\end{thm}

\begin{proof}
For each $i$, $1 \leq i \leq n$, $c(G_i) = c(G) + bl(v_i) - 1$. Now, applying sigma on both sides of the equation yields that $\sum_{i = 1}^nc(G_i)$ = $\sum_{i = 1}^nc(G)$ + $\sum_{i = 1}^nbl(v_i)$ $-$ $\sum_{i = 1}^n1$ which means that $\sum_{i = 1}^nc(H_i)$ = $nc(G)$ + $\sum_{i = 1}^nbl(u_i)$ $-$ $n$, by part (b) of Lemma \ref{cutnodes}. This shows that $nc(G) = nc(H)$ and thus, the number of connected components of $G$ and $H$ are the same.
\end{proof}

\vskip 3mm

In Theorem \ref{p}, we are showing that two hypomorphic graphs have the same number of paths of length $l$.

\vskip 3mm

\begin{thm}\label{p}
For two hypomorphic graphs $G$ and $H$, $p_G(v_j,l) = p_H(u_j,l)$ and $p_G(v_i, v_j, l) = p_H(u_i, u_j, l)$, where  $1 \leq i \neq j \leq n$ and $1 \leq l \leq n - 2$.
\end{thm}
\begin{proof}
Since for each $i$, $G_i \cong H_i$, $p_{G_i}(v_j , l) = p_{H_i}(u_j , l)$ in which $i \neq j$. It is clear to see that
\begin{enumerate}
\item[(2)]  $p_G(v_j , l) = p_{G_i}(v_j , l) + p_G(v_j , v_i , l),$
\item[(3)] $p_H(u_j , l) = p_{H_i}(u_j , l) + p_H(u_j , u_i , l).$
\end{enumerate}
Applying sigma on both sides of equals (2) yields that $\sum_{\stackrel{i = 1}{i \neq j}}^np_G(v_j , l)$ = $\sum_{\stackrel{i = 1}{i \neq j}}^np_{G_i}(v_j , l)$ + $\sum_{\stackrel{i = 1}{i \neq j}}^np_G(v_j , v_i , l)$. This implies that $(n - 1)p_G(v_j,l)$ = $\sum_{\stackrel{i = 1}{i \neq j}}^np_{G_i}(v_j,l)$ + $lp_G(v_j,l)$, by Lemma \ref{implem}. Therefore, $\sum_{\stackrel{i = 1}{i \neq j}}^np_{G_i}(v_j,l)$ = $(n - l - 1)p_G(v_j,l)$.
A similar argument as above and Equation (3) show that in $H$, $\sum_{\stackrel{i = 1}{i \neq j}}^np_{H_i}(u_j , l)$ = $(n - l - 1)p_H(u_j , l)$ which immediately leads to $p_G(v_j,l) = p_H(u_j,l)$, for $1 \leq l \leq n - 2$. We now apply (2) and (3) to prove that $p_G(v_i,v_j,l)$ = $p_H(u_i,u_j,l)$. This completes the proof.
\end{proof}

\vskip 3mm

The next corollary proves the Kelly-Ulam conjecture  in the category of simple and finite graphs.

\vskip 3mm

\begin{cor}
If $G \dot{\cong} H$, then $G \cong H$.
\end{cor}
\begin{proof}
Suppose $G$ and $H$ are two simple and finite hypomorphic graphs. By putting $l = 1$ in Theorem \ref{p}, it is observed that $p_G(v_i,v_j,1) = p_H(u_i, u_j, 1)$ for all $i, j$, $1 \leq i \neq j \leq n$. This means that $v_iv_j \in E(G)$ if and only if $u_iu_j \in E(H)$, which concludes that $G$ = $H$.
\end{proof}

\vskip 3mm

\noindent{\bf Acknowledgement.} The research of the second author is partially supported by the University of Kashan under grant no. 364988/66.