Twisted patterns in large subsets of ZN

Journal article, 2017

Let E ? ZN be a set of positive upper Banach density, and let ? < GLN (Z) be a "su ciently large" subgroup. We show in this paper that for each positive integer m there exists a positive integer k with the following property: For every fa {a1,?, am} ? k · ZN , there are ?1,?,?m ? ? and b ? E such that [formula presented] We use this “twisted” multiple recurrence result to study images of E - b under various -invariant maps. For instance, if N ? 3 and Q is an integer quadratic form on ZN of signature (p, q) with p, q ? 1 and p + q ? 3, then our twisted multiple recurrence theorem applied to the group ? = SO (Q) (Z) shows that [formula presented] for every F ? k · ZN with m elements. In the case when E is an aperiodic Bohro set, we can choose b to be zero and k = 1, and thus Q (ZN) ? Q (E). 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. © Swiss Mathematical Society.