Note on the pinned distance problem over finite fields

TL;DR

This paper investigates the pinned distance problem over finite fields, showing that large sets contain many points with rich distance structures, using averaging and pigeonhole principles to establish the existence of such points.

Contribution

It proves the existence of points with many distances to large sets in finite fields, extending the understanding of distance problems in finite geometry.

Findings

Existence of points with many distances in large finite field sets

Large sets determine a positive proportion of all distances when combined with certain points

Averaging and pigeonhole principles are key tools in the proofs

Abstract

Let F_q be a finite field with odd q elements. In this article, we prove that if E \subseteq \mathbb F_q^d, d\ge 2, and |E|\ge q, then there exists a set Y \subseteq \mathbb F_q^d with |Y|\sim q^d$ such that for all y\in Y, the number of distances between the point y and the set E is similar to the size of the finite field \mathbb F_q. As a corollary, we obtain that for each set E\subseteq \mathbb F_q^d with |E|\ge q, there exists a set Y\subseteq \mathbb F_q^d with |Y|\sim q^d so that any set E\cup \{y\} with y\in Y determines a positive proportion of all possible distances. An averaging argument and the pigeonhole principle play a crucial role in proving our results.