Diophantine patterns via non-random walks

Abstract

In previous works of Bj{\"o}rklund, Fish and myself, it was shown that if \(B \subset \mathbb{Z}^d\) has positive density, then \(Q(B-B)\) contains a subgroup \(k \mathbb{Z}\), where \(Q:\mathbb{Z}^d \to \mathbb{Z}\) is a homogeneous polynomial mapping with sufficiently large symmetry group in \(\operatorname{SL}_d(\mathbb{Z})\), such as a non-positive definite quadratic form. We give a much more elementary proof (the previous ones used recent deep results of Benoist-Quint on random walks) of this result for many (but not all) such \(Q\), as well as some new non-homogeneous examples. The idea is to exploit the existence of polynomial walks, rather than random walks, in these symmetry groups and apply the classical polynomial equidistribution result of Weyl, instead of the much more difficult random walk equidistribution results of Benoist-Quint. No background in Ergodic Theory will be assumed.