  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 $\epsilon$.
