Averages of point configuration problems over finite p-adic rings.

This paper studies averages of finite point configuration problems for subsets E ⊂ (Z/p^r)ⁿ (r ≥ 1, n ≥ 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. [ABSTRACT FROM AUTHOR]