JournalscmhVol. 92, No. 3pp. 621–640

Twisted patterns in large subsets of ZN\mathbb Z^N

  • Michael Björklund

    Chalmers University of Technology, Gothenburg, Sweden
  • Kamil Bulinski

    University of Sydney, Australia
Twisted patterns in large subsets of $\mathbb Z^N$ cover
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Let EZNE \subset \mathbb Z^N be a set of positive upper Banach density, and let Γ<GLN(Z)\Gamma < \mathrm {GL}_N(\mathbb Z) be a "sufficiently large" subgroup. We show in this paper that for each positive integer mm there exists a positive integer kk with the following property: For every {a1,,am}kZN\{a_1,\ldots,a_m\} \subset k \cdot \mathbb Z^N, there are γ1,,γmΓ\gamma_1,\ldots,\gamma_m \in \Gamma and bEb \in E such that

γiaiEb,for all i=1,,m.\gamma_i \cdot a_i \in E - b, \quad \text{for all $i = 1,\ldots,m$}.

We use this „twisted" multiple recurrence result to study images of EbE-b under various Γ\Gamma-invariant maps. For instance, if N3N \geq 3 and QQ is an integer quadratic form on ZN\mathbb Z^N of signature (p,q)(p,q) with p,q1p,q \geq 1 and p+q3p + q \geq 3, then our twisted multiple recurrence theorem applied to the group Γ=SO(Q)(Z)\Gamma = \mathrm {SO}(Q)(\mathbb Z) shows that

k2Q(F)Q(Eb),k^2 Q(F) \subset Q(E-b),

for every FkZNF \subset k \cdot \mathbb Z^N with mm elements. In the case when EE is an aperiodic Bohro_o set, we can choose bb to be zero and k=1k = 1, and thus Q(ZN)Q(E)Q(\mathbb Z^N) \subset 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.

Cite this article

Michael Björklund, Kamil Bulinski, Twisted patterns in large subsets of ZN\mathbb Z^N. Comment. Math. Helv. 92 (2017), no. 3, pp. 621–640

DOI 10.4171/CMH/420