Twisted recurrence via polynomial walks

Abstract: 

We will show how polynomial walks can be used to establish a twisted recurrence for sets of positive density in ZdZd. In particular, we will demonstrate that if Γ≤GLd(Z)Γ≤GLd(Z) is finitely generated by unipotents and acts irreducibly on RdRd, then for any set B⊂ZB⊂Z d of positive density, there exists k≥1k≥1 such that for any v∈kZdv∈kZd one can find γ∈Γγ∈Γ with γv∈B−Bγv∈B−B. Also we will show a non-linear analog of Bogolubov’s theorem – for any set B⊂Z2B⊂Z2 of positive density, and p(n)∈Z[n]p(n)∈Z[n], p(0)=0p(0)=0, degp≥2deg⁡p≥2, there exists k≥1k≥1 such that kZ⊂{x−p(y)|(x,y)∈B−B}kZ⊂{x−p(y)|(x,y)∈B−B}. Joint work with Kamil Bulinski.