Averages of point configuration problems over finite 𝑝-adic rings

Lichtin, Ben

doi:10.1090/proc/15449

Averages of point configuration problems over finite $p-$adic rings
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Proc. Amer. Math. Soc. 149 (2021), 2825-2839 Request permission

Abstract:

This paper studies averages of finite point configuration problems for subsets $E \subset (\mathbb {Z}/p^{r})^n$ ($r \ge 1, n \ge 2$) and extends work of Bennett-Hart-Iosevich-Pakianathan-Rudnev over finite fields to finite $p-$adic rings. As a result, we show that averages, taken over the group of orthogonal transformations, of finite point configurations with endpoints in $E$ are positive if the density of $E$ is sufficiently large.

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