Results on the Erd\H os-Falconer distance problem in $\mathbb{Z}_q^d$ for odd $q$

TL;DR

This paper extends the Erdős-Falconer distance problem results in $Z_q^d$ to all odd $q$, proving that large sets have distance sets covering entire $Z_q$, not just units, thus generalizing previous prime power cases.

Contribution

It generalizes the Erdős-Falconer distance problem to all odd integers $q$, removing the prime power restriction and showing large sets' distances cover all of $Z_q$.

Findings

Distance set of large $E$ equals $Z_q$ for odd $q$

Results extend previous prime power cases to all odd $q$

Distance sets contain all elements of $Z_q$ for sufficiently large $E$

Abstract

The Erd\H os-Falconer distance problem in $\mathbb{Z}_q^d$ asks one to show that if $E \subset \mathbb{Z}_q^d$ is of sufficiently large cardinality, then $\Delta(E) := \{(x_1 - y_1)^2 + \dots + (x_d - y_d)^2 : x, y \in E\}$ satisfies $\Delta(E) = \mathbb{Z}_q$. Here, $\mathbb{Z}_q$ is the set of integers modulo $q$, and $\mathbb{Z}_q^d$ is the free module of rank $d$ over $\mathbb{Z}_q$. We extend known results in two directions. Previous results were known only in the setting $q = p^{\ell}$, where $p$ is an odd prime, and as such only showed that all units were obtained in the distance set. We remove the constriction that $q$ is a power of a prime, and despite this, shows that the distance set of $E$ contains \emph{all} of $\mathbb{Z}_q$ whenever $E$ is sufficiently large.