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Given an arbitrary $1$-Lipschitz function $f$ on the torus $\mathbb{T}^n $,we find a $k$-dimensional subtorus $M \subseteq \mathbb{T}^n$, parallel to theaxes, such that the restriction of $f$ to the subtorus $M$ is nearly a constantfunction. The $k$-dimensional subtorus $M$ is chosen randomly and uniformly. Weshow that when $k \leq c \log n / (\log \log n + \log 1/\varepsilon)$, themaximum and the minimum of $f$ on this random subtorus $M$ differ by at most$\varepsilon$, with high probability.

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