Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Pythagorean powers of hypercubes   For $n\in \mathbb{N}$ consider the $n$-dimensional hypercube as equal to the
vector space $\mathbb{F}_2^n$, where $\mathbb{F}_2$ is the field of size two.
Endow $\mathbb{F}_2^n$ with the Hamming metric, i.e., with the metric induced
by the $\ell_1^n$ norm when one identifies $\mathbb{F}_2^n$ with
$\{0,1\}^n\subseteq \mathbb{R}^n$. Denote by $\ell_2^n(\mathbb{F}_2^n)$ the
$n$-fold Pythagorean product of $\mathbb{F}_2^n$, i.e., the space of all
$x=(x_1,\ldots,x_n)\in \prod_{j=1}^n \mathbb{F}_2^n$, equipped with the metric
$$ \forall\, x,y\in \prod_{j=1}^n \mathbb{F}_2^n,\qquad
d_{\ell_2^n(\mathbb{F}_2^n)}(x,y)= \sqrt{
\|x_1-y_1\|_1^2+\ldots+\|x_n-y_n\|_1^2}. $$ It is shown here that the
bi-Lipschitz distortion of any embedding of $\ell_2^n(\mathbb{F}_2^n)$ into
$L_1$ is at least a constant multiple of $\sqrt{n}$. This is achieved through
the following new bi-Lipschitz invariant, which is a metric version of (a
slight variant of) a linear inequality of Kwapie{\'n} and Sch\"utt (1989).
Letting $\{e_{jk}\}_{j,k\in \{1,\ldots,n\}}$ denote the standard basis of the
space of all $n$ by $n$ matrices $M_n(\mathbb{F}_2)$, say that a metric space
$(X,d_X)$ is a KS space if there exists $C=C(X)>0$ such that for every $n\in
2\mathbb{N}$, every mapping $f:M_n(\mathbb{F}_2)\to X$ satisfies
\begin{equation*}\label{eq:metric KS abstract}
\frac{1}{n}\sum_{j=1}^n\mathbb{E}\left[d_X\Big(f\Big(x+\sum_{k=1}^ne_{jk}\Big),f(x)\Big)\right]\le
C \mathbb{E}\left[d_X\Big(f\Big(x+\sum_{j=1}^ne_{jk_j}\Big),f(x)\Big)\right],
\end{equation*} where the expectations above are with respect to $x\in
M_n(\mathbb{F}_2)$ and $k=(k_1,\ldots,k_n)\in \{1,\ldots,n\}^n$ chosen
uniformly at random. It is shown here that $L_1$ is a KS space (with $C=
2e^2/(e^2-1)$, which is best possible), implying the above nonembeddability
statement. Links to the Ribe program are discussed, as well as related open
problems.

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The categories of this paper are: chao-dyn
34
The categories of this paper are: chem-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: dg-ga
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The categories of this paper are: adap-org
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The categories of this paper are: eess
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The categories of this paper are: cmp-lg
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The categories of this paper are: solv-int
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The categories of this paper are: nlin
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The categories of this paper are: physics
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The categories of this paper are: patt-sol
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The categories of this paper are: q-bio
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The categories of this paper are: acc-phys
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The categories of this paper are: stat
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The categories of this paper are: nucl-ex
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The categories of this paper are: math-ph