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

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 $\mathrm{\Delta }(E):=\{(x_1-y_1{)}^2+\cdots +(x_d-y_d{)}^2:x,y\in E\}$ satisfies $\mathrm{\Delta }(E)={\mathbb{Z}}_q$. 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 $q=p^{\mathrm{\ell }}$, 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.