The quotient set of the quadratic distance set over finite fields

Let F-q(d) be the d-dimensional vector space over the finite field F-q with q elements. For each non-zero r in F-q and E subset of F-q(d), we define W(r) as the number of quadruples (x,y, z, w) is an element of E-4 such that Q(x-y)/Q(z-w) = r, where Q is a non-degenerate quadratic form in d variables over F-q. When Q(alpha) = & sum;(d)(i =1) alpha(2)(i) with alpha = (alpha(1), ... , alpha(d)) is an element of F-q(d), Pham (2022) recently used the machinery of group actions and proved that if E subset of F-q(2) with q equivalent to 3 (mod 4) and |E| >= Cq, then we have W(r) >= c|E|(4)/q for any non-zero square number r is an element of 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.