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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-degeneratequadratic form in $d$ variables over $\mathbb F_q.$ When$Q(\alpha)=\sum_{i=1}^d \alpha_i^2$ with $\alpha=(\alpha_1, \ldots,\alpha_d)\in \mathbb F_q^d,$ Pham (2022) recently used the machinery of groupactions 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 squarenumber $r \in \mathbb F_q,$ where $C$ is a sufficiently large constant, $ c$ issome 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 toarbitrary dimensions with general non-degenerate quadratic distances. As acorollary of our results, we also generalize the sharp results on the Falconertype problem for the quotient set of distance set due to the first two authorsand Parshall (2019). Furthermore, we provide improved constants for the sizeconditions of the underlying sets.  The key new ingredient is to relate the estimate of the $W(r)$ to a quadratichomogeneous variety in $2d$-dimensional vector space. This approach is fruitfulbecause it allows us to take advantage of Gauss sums which are more handleablethan the Kloosterman sums appearing in the standard distance type problems.

The quotient set of the quadratic distance set over finite fields