# Pinned Distances in Modules over Finite Valuation Rings

## TL;DR

This paper establishes lower bounds on the number of distinct distances from a fixed point in the Cartesian product of a subset of a finite valuation ring, showing that large subsets determine many distances.

## Contribution

It proves a new lower bound on pinned distances in modules over finite valuation rings, extending distance problems to this algebraic setting.

## Key findings

- Existence of a point with many pinned distances
- If |A| ≥ q^{r - 1/3}, A×A determines a positive proportion of all distances
- Lower bounds depend on the size of A and the ring's parameters

## Abstract

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 $$|\Delta_u(A\times A)|\gg \min\left\{q^r, \frac{|A|^3}{q^{2r-1}}\right\}.$$ 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.