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

Given an arbitrary 1-Lipschitz function f on the torus Tⁿ, we find a k-dimensional subtorus M ⊆ Tⁿ, 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 selected randomly and uniformly. We show that when k ≤ c log n/(log log n + log 1/ε), the maximum and the minimum of f on this random subtorus M differ by at most ε, with high probability.