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arXiv · 1311.6958

Juntas in the $\ell^{1}$-grid and Lipschitz maps between discrete tori

Itai Benjamini·David Ellis·Ehud Friedgut·Nathan Keller·Arnab Sen
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Abstract

We show that if $A \subset [k]^n$, then $A$ is $\epsilon$-close to a junta depending upon at most $\exp(O(|\partial A|/(k^{n-1}\epsilon)))$ coordinates, where $\partial A$ denotes the edge-boundary of $A$ in the $\ell^1$-grid. This is sharp up to the value of the absolute constant in the exponent. This result can be seen as a generalisation of the Junta theorem for the discrete cube, from [E. Friedgut, Boolean functions with low average sensitivity depend on few coordinates, Combinatorica 18 (1998), 27-35], or as a characterization of large subsets of the $\ell^1$-grid whose edge-boundary is small. We use it to prove a result on the structure of Lipschitz functions between two discrete tori; this can be seen as a discrete, quantitative analogue of a recent result of Austin [T. Austin, On the failure of concentration for the $\ell^{\infty}$-ball, preprint]. We also prove a refined version of our junta theorem, which is sharp in a wider range of cases.

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Itai Benjamini, David Ellis, Ehud Friedgut, Nathan Keller, Arnab Sen. 2013-11-27. Juntas in the $\ell^{1}$-grid and Lipschitz maps between discrete tori. https://arxiv.org/abs/1311.6958

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