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

TL;DR

This paper generalizes the Junta theorem to subsets of the $ ext{l}^1$-grid, characterizes their structure via edge-boundary size, and applies this to analyze Lipschitz functions between discrete tori, providing sharp bounds and a refined version.

Contribution

It extends the Junta theorem to the $ ext{l}^1$-grid, linking edge-boundary size to junta approximation, and applies this to Lipschitz maps between discrete tori, with sharp bounds and refinements.

Findings

A subset of the $ ext{l}^1$-grid is close to a junta depending on boundary size.

The result is sharp up to a constant in the exponent.

Application to Lipschitz functions between discrete tori with a discrete analogue of Austin's result.

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.