On the oscillation rigidity of a Lipschitz function on a high-dimensional flat torus

Abstract

Given an arbitrary $1$-Lipschitz function $f$ on the torus ${\mathbb{T}}^n$, we find a $k$-dimensional subtorus $M\subseteq {\mathbb{T}}^n$, parallel to the axes, such that the restriction of $f$ to the subtorus $M$ is nearly a constant function. The $k$-dimensional subtorus $M$ is chosen randomly and uniformly. We show that when $k\le c\log n/(\log \log n+\log 1/\epsilon )$, the maximum and the minimum of $f$ on this random subtorus $M$ differ by at most $\epsilon$, with high probability.

8 pages