Question

I believe that Hillel Furstenberg uses the von Neumann ergodic theorem quite frequently in his work on recurrence, which has applications to number theory. For example, in section 3 of the article Poincaré recurrence and number theory he uses Weyl's criterion and the von Neumann ergodic theorem to prove the following result: if $T$ is a measure-preserving transformation of a probability space, $p$ is a polynomial with integer coefficients and no constant term, and $A$ is a positive-measure subset of the probability space in question, then there are infinitely many natural numbers $t$ such that $T^{-p(t)}A \cap A$ has postive measure. A corollary of this result is that if $X$ is a subset of the integers with positive density and $p$ is an integer polynomial with no constant term, then the equation $x-y=p(t)$ can be solved for $x,y \in X$ and $t$ a positive integer. The von Neumann ergodic theorem is also used in ergodic proofs of Roth's theorem (see for example the exposition by Á. Magyar). There are probably more examples in the book Recurrence in Ergodic Theory and Combinatorial Number Theory.