Group action approaches in Erdős quotient set problem

Let $F_q$ denote the finite field of q elements. Let $F_q^d$ be the d-dimensional vector space over the field $F_q$. Let $F_q^{⁎}:=F_q\setminus \left\{0\right\}$. Let ${\left(F_q\right)}^2:=\{t^2:t\in F_q\}$. For $E\subset F_q^d$, denote the distance set by $\Delta \left(E\right):=\{\Vert x-y\Vert :={(x_1-y_1)}^2+\cdots +{(x_d-y_d)}^2:x,y\in E\}$. Denote the Erdős quotient set by $\frac{\Delta \left(E\right)}{\Delta \left(E\right)}:=\{\frac{s}{t}:s,t\in \Delta (E),t\ne 0\}=F_q$.

The Erdős quotient set problem was introduced in [13] where it was shown that for even $d\ge 2$, if $E\subset F_q^2$ such that $\left|E\right|\gg q^{d/2}$, then $\frac{\Delta \left(E\right)}{\Delta \left(E\right)}=F_q$. The proof of the latter result is quite sophisticated, and in [17] a simple proof using a group-action approach was obtained for the case of $q\equiv 3\left(\mathrm{mod}4\right)$ and $d=2$. In the $q\equiv 3\left(\mathrm{mod}4\right)$ setting, for each $r\in {\left(F_q\right)}^2$, [17] showed if $E\subset F_q^2$, then $V\left(r\right):=\#\left\{\right(a,b,c,d)\in E^4:\frac{\Vert a-b\Vert }{\Vert c-d\Vert }=r\}\gg \frac{|E{|}^4}{q}$.

In this work we use group action techniques in the $q\equiv 3\left(\mathrm{mod}4\right)$ setting for $d=2$ and improve the results of [17] by removing the assumption on $r\in {\left(F_q\right)}^2$. Specifically we show if $d=2$, $q\equiv 3\left(\mathrm{mod}4\right)$, and $\left|E\right|\ge \sqrt{2}q$, then for each $r\in F_q^{⁎}$ it follows that $V\left(r\right)\ge \frac{|E{|}^4}{2q}$. Finally, we improve the main result of [2] using the proof techniques from our quotient set results. Our novel introduction of the matrices $A_{\text{even}}$ and $A_{\text{odd}}$ is the key to improving the results of [17], [2].