- journal article metadata
European Mathematical Society Publishing House
2017-07-30 23:45:01
Commentarii Mathematici Helvetici
Comment. Math. Helv.
CMH
0010-2571
1420-8946
General
10.4171/CMH
http://www.ems-ph.org/doi/10.4171/CMH
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European Mathematical Society Publishing House
Zuerich, Switzerland
© Swiss Mathematical Society
92
2017
3
Twisted patterns in large subsets of $\mathbb Z^N$
Michael
Björklund
Chalmers University of Technology, Gothenburg, Sweden
Kamil
Bulinski
University of Sydney, Australia
Multiple recurrence, equidistribution, invariants
Let $E \subset \mathbb Z^N$ be a set of positive upper Banach density, and let $\Gamma < \mathrm {GL}_N(\mathbb Z)$ be a "sufficiently 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 $\{a_1,\ldots,a_m\} \subset k \cdot \mathbb Z^N$, there are $\gamma_1,\ldots,\gamma_m \in \Gamma$ and $b \in E$ such that $$\gamma_i \cdot a_i \in E - b, \quad \text{for all $i = 1,\ldots,m$}.$$ We use this „twisted" multiple recurrence result to study images of $E-b$ under various $\Gamma$-invariant maps. For instance, if $N \geq 3$ and $Q$ is an integer quadratic form on $\mathbb Z^N$ of signature $(p,q)$ with $p,q \geq 1$ and $p + q \geq 3$, then our twisted multiple recurrence theorem applied to the group $\Gamma = \mathrm {SO}(Q)(\mathbb Z)$ shows that $$k^2 Q(F) \subset Q(E-b),$$ for every $F \subset k \cdot \mathbb Z^N$ with $m$ elements. In the case when $E$ is an aperiodic Bohr$_o$ set, we can choose $b$ to be zero and $k = 1$, and thus $Q(\mathbb Z^N) \subset Q(E)$. 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.
Dynamical systems and ergodic theory
Number theory
621
640
10.4171/CMH/420
http://www.ems-ph.org/doi/10.4171/CMH/420