Pinned Distances in Modules over Finite Valuation Rings

Let $R$ be a finite valuation ring of order $q^r$ where $q$ is odd and $A$ be a subset of $R$. In the present paper, we prove that there exists a point $u$ in the Cartesian product set $A\times A\subset R^2$ such that the size of the pinned distance set at $u$ satisfies $|Δ_u(A\times A)|\gg \min\left\{q^r, \frac{|A|^3}{q^{2r-1}}\right\}.$ This implies that if $|A|\ge q^{r-\frac{1}{3}}$, then the set $A\times A$ determines a positive proportion of all possible distances.

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