Lipschitz functions on the infinite-dimensional torus

We discuss the spectrum phenomenon for Lipschitz functions on the infinite-dimensional torus. Suppose that [Formula: see text] is a measurable, real-valued, Lipschitz function on the torus [Formula: see text]. We prove that there exists a number [Formula: see text] with the following property: For any [Formula: see text], there exists a parallel, infinite-dimensional subtorus [Formula: see text] such that the restriction of the function [Formula: see text] to the subtorus [Formula: see text] has an [Formula: see text]-norm of at most [Formula: see text].