2017/06/24 by Bulinski, Kamil, Fish, Alexander
#11B05 #11B30 #37A05 #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1706.07921
In this paper we show how polynomial walks can be used to establish a twisted recurrence for sets of positive density in ℤd. In particular, we prove that if Γ≤ GLd(ℤ) is finitely generated by unipotents and acts irreducibly on ℝd, then for any set B ⊂ ℤd of positive density, there exists k ≥ 1 such that for any v ∈ k ℤd one can find γ∈ Γ with γv ∈ B - B. Our method does not require the linearity of the action, and we prove a twisted recurrence for semigroups of maps from ℤd to ℤd satisfying some irreducibility and polynomial assumptions. As one of the consequences, we prove a non-linear analog of Bogolubov's theorem -- for any set B ⊂ ℤ2 of positive density, and p(n) ∈ ℤ[n], with p(0) = 0 and deg(p) ≥ 2, there exists k ≥ 1 such that k ℤ ⊂ \ x - p(y) | (x,y) ∈ B-B \. Unlike the previous works on twisted recurrence that used recent results of Benoist-Quint and Bourgain-Furman-Lindenstrauss-Mozes on equidistribution of random walks on automorphism groups of tori, our method relies on the classical Weyl equidistribution for polynomial orbits on tori.