Twisted recurrence via polynomial walks


Alexander Fish


University of Sydney


Wed, 30/08/2017 - 2:00pm


RC-4082, The Red Centre, UNSW


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

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