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arXiv · 1706.10010

IP$^{*}$-sets in function field and mixing properties

Dibyendu De·Pintu Debnath
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Abstract

The ring of polynomial over a finite field $F_q[x]$ has received much attention, both from a combinatorial viewpoint as in regards to its action on measurable dynamical systems. In the case of $(\mathbb{Z},+)$ we know that the ideal generated by any nonzero element is an IP$^*$-set. In the present article we first establish that the analogous result is true for $F_q[x]$. We further use this result to establish some mixing properties of the action of $(F_q[x],+)$. We shall also discuss on Khintchine's recurrence for the action of $(F_q[x]\setminus\{0\},\cdot)$.

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Dibyendu De, Pintu Debnath. 2017-06-30. IP$^{*}$-sets in function field and mixing properties. https://arxiv.org/abs/1706.10010

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