Group Action Approaches in Erdos Quotient Set Problem

Let $\mathbb{F}_q$ denote the finite field of $q$ elements. For $E \subset \mathbb{F}_q^d$, denote the distance set $\Delta(E)= \{\|x-y\|^2:=(x_1-y_1)^2+ \cdots + (x_d-y_d)^2 : (x,y)\in E^2 \}$. The Erdos quotient set problem was introduced in \cite{Iosevich_2019} where it was shown that for even $d\geq2$ that if $|E| \subset \mathbb{F}_q^2$ such that $|E| >> q^{d/2}$, then $\frac{\Delta(E)}{\Delta(E)}:= \{\frac{s}{t}:s,t \in \Delta(E), t\not=0\} =\mathbb{F}_q^d$. The proof of the latter result is quite sophisticated and in \cite{pham2023group}, a simple proof using a group-action approach was obtained for the case of $q \equiv 3 \mod 4$ when $d=2$. In the $q \equiv 3 \mod 4$ setting, for each $r \in (\mathbb{F}_q)^2$, \cite{pham2023group} showed if $E \subset \mathbb{F}_q$, then $V(r):= \# \left\{ (a,b,c,d) \in E^2: \frac{\|a-b\|^2}{\|c-d\|^2} = r \right\} >> \frac{|E|^4}{q}$. In this work we use group action techniques in the $q \equiv 3 \mod 4$ setting, for $d=2$ and improve the results of \cite{pham2023group} by removing the assumption on $r \in (\mathbb{F}_q)^2$. Specifically we show if $d=2$ and $q \equiv 3 \mod 4$, then for each $r \in \mathbb{F}_q^*$,$V(r)\geq \frac{|E|^4}{2q}$if $|E|\geq \sqrt{2}q$ for all $r \in \mathbb{F}_q$. Finally, we improve the main result of \cite{bhowmik2023near} using our proof techniques from our quotient set results.