A new bound on Erd\H{o}s distinct distances problem in the plane over prime fields

TL;DR

This paper establishes a new lower bound on the number of distinct distances determined by point sets in the plane over prime fields, improving previous bounds through advanced combinatorial and incidence geometry techniques.

Contribution

It introduces a novel lower bound for the Erd ext{o}s distinct distances problem over prime fields, extending the known range of set sizes where the bound applies.

Findings

New lower bound for |Δ(A)| when |A| ≤ p^{7/6}

Improved bounds for |A| ≤ p^{1+149/4065}

Utilization of energy on paraboloids and incidence bounds to achieve results

Abstract

In this paper we obtain a new lower bound on the Erd\H{o}s distinct distances problem in the plane over prime fields. More precisely, we show that for any set $A\subset \mathbb{F}_p^2$ with $|A|\le p^{7/6}$, the number of distinct distances determined by pairs of points in $A$ satisfies $$ |\Delta(A)| \gg |A|^{\frac{1}{2}+\frac{149}{4214}}.$$ Our result gives a new lower bound of $|\Delta{(A)}|$ in the range $|A|\le p^{1+\frac{149}{4065}}$. The main tools we employ are the energy of a set on a paraboloid due to Rudnev and Shkredov, a point-line incidence bound given by Stevens and de Zeeuw, and a lower bound on the number of distinct distances between a line and a set in $\mathbb{F}_p^2$. The latter is the new feature that allows us to improve the previous bound due Stevens and de Zeeuw.