Lipschitz functions on the infinite-dimensional torus

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 ∞. We prove that there exists a number a R with the following property: For any ε > 0, there exists a parallel, infinite-dimensional subtorus M ⊆ ∞ such that the restriction of the function f - a to the subtorus M has an L∞(M)-norm of at most ε.