Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existence of similar point configurations in thin subsets of $\Bbb R^d$   We prove the existence of similar and multi-similar point configurations (or
simplexes) in sets of fractional Hausdorff measure in Euclidean space. These
results can be viewed as variants, for thin sets, of theorems for sets of
positive density in $\Bbb R^d$ due to Furstenberg, Katznelson and Weiss
\cite{FKW90}, Bourgain \cite{B86} and Ziegler \cite{Z06}.
  Let $d \ge 2$ and $E\subset {\Bbb R}^d$ be a compact set. For $k\ge 1$,
define $$\Delta_k(E)=\left\{\left(|x^1-x^2|, \dots, |x^i-x^j|,\dots,
|x^k-x^{k+1}|\right): \left\{x^i\right\}_{i=1}^{k+1}\subset E\right\} \subset
{\Bbb R}^{k(k+1)/2}, $$ the {\it $(k+1)$-point configuration set} of $E$. For
$k\le d$, this is (up to permutations) the set of congruences of $(k+1)$-point
configurations in $E$; for $k>d$, it is the edge-length set of $(k+1)$-graphs
whose vertices are in $E$. Previous works by a number of authors have found
values $s_{k,d}<d$ so that if the Hausdorff dimension of $E$ is $>s_{k,d}$,
then $\Delta_k(E)$ has positive Lebesgue measure. In this paper we study more
refined properties of $\Delta_k(E)$, namely the existence of (exactly) similar
or multi--similar configurations. For $r\in\Bbb R,\, r>0$, let
$$\Delta_{k}^{r}(E):=\left\{\vec{t}\in \Delta_k\left(E\right): r\vec{t}\in
\Delta_k\left(E\right)\right\}\subset \Delta_k\left(E\right).$$ We show that
for all $E$ with Hausdorff dimension $>s_{k,d}$, a natural measure $\nu_k$ on
$\Delta_k(E)$ and all $r\in\Bbb R_+$, one has
$\nu_k\left(\Delta_{k}^{r}\left(E\right)\right)>0$. Thus, there exist many
pairs, $\{x^1, x^2, \dots, x^{k+1}\}$ and $\{y^1, y^2, \dots, y^{k+1}\}$, in
$E$ which are similar by the scaling factor $r$.
  We also show the existence of triply-similar and multi-similar
configurations.

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The categories of this paper are: hep-ex
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The categories of this paper are: q-bio
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The categories of this paper are: stat
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The categories of this paper are: bayes-an
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The categories of this paper are: supr-con
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The categories of this paper are: mtrl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: chem-ph
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The categories of this paper are: q-fin
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The categories of this paper are: solv-int
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The categories of this paper are: cond-mat
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The categories of this paper are: adap-org
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The categories of this paper are: astro-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: econ
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The categories of this paper are: math-ph