The quotient set of the quadratic distance set over finite fields

Abstract

Let $\mathbb F_q^d$ be the $d$-dimensional vector space over the finite field $\mathbb F_q$ with $q$ elements. For each non-zero $r$ in $\mathbb F_q$ and $E\subset \mathbb F_q^d$, we define $W(r)$ as the number of quadruples $(x,y,z,w)\in E^4$ such that $ Q(x-y)/Q(z-w)=r,$ where $Q$ is a non-degenerate quadratic form in $d$ variables over $\mathbb F_q.$ When $Q(α)=\sum_{i=1}^d α_i^2$ with $α=(α_1, \ldots, α_d)\in \mathbb F_q^d,$ Pham (2022) recently used the machinery of group actions and proved that if $E\subset \mathbb F_q^2$ with $q\equiv 3 \pmod{4}$ and $|E|\ge C q$, then we have $W(r)\ge c |E|^4/q$ for any non-zero square number $r \in \mathbb F_q,$ where $C$ is a sufficiently large constant, $ c$ is some number between $0$ and $1,$ and $|E|$ denotes the cardinality of the set $E.$ In this article, we improve and extend Pham's result in two dimensions to arbitrary dimensions with general non-degenerate quadratic distances. As a corollary of our results, we also generalize the sharp results on the Falconer type problem for the quotient set of distance set due to the first two authors and Parshall (2019). Furthermore, we provide improved constants for the size conditions of the underlying sets. The key new ingredient is to relate the estimate of the $W(r)$ to a quadratic homogeneous variety in $2d$-dimensional vector space. This approach is fruitful because it allows us to take advantage of Gauss sums which are more handleable than the Kloosterman sums appearing in the standard distance type problems.