Pinned algebraic distances determined by Cartesian products in $\mathbb{F}_p^2$

TL;DR

This paper proves that for subsets of finite fields, the number of distinct pinned algebraic distances determined by Cartesian products is at least proportional to the minimum of the field size and the subset size raised to the 3/2 power.

Contribution

It establishes a new lower bound on the number of pinned algebraic distances in finite fields for Cartesian product sets.

Findings

At least a constant multiple of min{p, |A|^{3/2}} distances are determined.

The result applies to subsets of finite fields with prime order.

Provides a quantitative measure of distance diversity in finite field Cartesian products.

Abstract

Let $p$ be an odd prime and $A \subseteq \mathbb{F}_p$ be a subset of the finite field with $p$ elements. We show that $A \times A \subseteq \mathbb{F}_p^2$ determines at least a constant multiple of $\min\{p, |A|^{3/2}\}$ distinct pinned algebraic distances.