Group Action Approaches in Erdos Quotient Set Problem

Let Fq denote the finite field of q elements. For E ⊂ Fqᵈ, denote the distance set Δ(E)= \\|x-y\|²:=(x₁-y₁)²+ ⋯ + (xd-yd)² : (x,y)∈ E² \. The Erdos quotient set problem was introduced in {Iosevich_2019} where it was shown that for even d≥2 that if |E| ⊂ Fq² such that |E| >> qd/2, then Δ(E)/Δ(E):= /t:s,t ∈ Δ(E), t=0\ =Fqᵈ. The proof of the latter result is quite sophisticated and in {pham2023group}, a simple proof using a group-action approach was obtained for the case of q ≡ 3 4 when $d=2$. In the q ≡ 3 4 setting, for each r ∈ (Fq)², {pham2023group} showed if E ⊂ Fq, then V(r):= # \ (a,b,c,d) ∈ E²: \|a-b\|²/\|c-d\|² = r \ >> |E|⁴/q. In this work we use group action techniques in the q ≡ 3 4 setting, for $d=2$ and improve the results of {pham2023group} by removing the assumption on r ∈ (Fq)². Specifically we show if $d=2$ and q ≡ 3 4, then for each r ∈ Fq^*,V(r)≥ |E|⁴/2qif |E|≥ √2q for all r ∈ Fq. Finally, we improve the main result of {bhowmik2023near} using our proof techniques from our quotient set results.