Lipschitz functions on the infinite-dimensional torus

Abstract

We discuss the spectrum phenomenon for Lipschitz functions on the infinite-dimensional torus. Suppose that $f$ is a measurable, real-valued, Lipschitz function on the torus ${\mathbb{T}}^{\infty }$. We prove that there exists a number $a\in \mathbb{R}$ with the following property: For any $\epsilon >0$ there exists a parallel, infinite-dimensional subtorus $M\subseteq {\mathbb{T}}^{\infty }$ such that the restriction of the function $f-a$ to the subtorus $M$ has an $L^{\infty }(M)$-norm of at most $\epsilon$.