Lipschitz functions on the infinite-dimensional torus

TL;DR

This paper investigates the spectral properties of Lipschitz functions on the infinite-dimensional torus, showing that such functions can be approximated uniformly on infinite-dimensional subtori by constants.

Contribution

It proves the existence of a constant and infinite-dimensional subtori where Lipschitz functions are uniformly close to that constant, revealing a spectral phenomenon in infinite dimensions.

Findings

Existence of a real number approximating Lipschitz functions on subtori

Construction of infinite-dimensional subtori with controlled function oscillation

Extension of spectral analysis to infinite-dimensional tori

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 $\epsilon$.