Let be a set of positive upper Banach density, and let be a "sufficiently large" subgroup. We show in this paper that for each positive integer there exists a positive integer with the following property: For every , there are and such that
We use this „twisted" multiple recurrence result to study images of under various -invariant maps. For instance, if and is an integer quadratic form on of signature with and , then our twisted multiple recurrence theorem applied to the group shows that
for every with elements. In the case when is an aperiodic Bohr set, we can choose to be zero and , and thus . Our result is derived from a non-conventional ergodic theorem which should be of independent interest. Important ingredients in our proofs are the recent breakthroughs by Benoist–Quint and Bourgain–Furman–Lindenstrauss–Mozes on equidistribution of random walks on automorphism groups of tori.
Cite this article
Michael Björklund, Kamil Bulinski, Twisted patterns in large subsets of . Comment. Math. Helv. 92 (2017), no. 3, pp. 621–640DOI 10.4171/CMH/420