The quotient set of the quadratic distance set over finite fields

Let 𝔽 q d {\mathbb{F}_{q}^{d}} be the d-dimensional vector space over the finite field 𝔽 q {\mathbb{F}_{q}} with q elements. For each non-zero r in 𝔽 q {\mathbb{F}_{q}} and E ⊂ 𝔽 q d {E\subset\mathbb{F}_{q}^{d}} , we define W ⁢ ( r ) {W(r)} as the number of quadruples ( x , y , z , w ) ∈ E 4 {(x,y,z,w)\in E^{4}} such that Q ⁢ ( x - y ) Q ⁢ ( z - w ) = r {\frac{Q(x-y)}{Q(z-w)}=r} , where Q is a non-degenerate quadratic form in d variables over 𝔽 q {\mathbb{F}_{q}} . When Q ⁢ ( α ) = ∑ i = 1 d α i 2 {Q(\alpha)=\sum_{i=1}^{d}\alpha_{i}^{2}} with α = ( α 1 , … , α d ) ∈ 𝔽 q d {\alpha=(\alpha_{1},\ldots,\alpha_{d})\in\mathbb{F}_{q}^{d}} , Pham (2022) recently used the machinery of group actions and proved that if E ⊂ 𝔽 q 2 {E\subset\mathbb{F}_{q}^{2}} with q ≡ 3 ⁢ ( mod ⁡ 4 ) {q\equiv 3~{}(\operatorname{mod}\,4)} and | E | ≥ C ⁢ q {|E|\geq Cq} , then we have W ⁢ ( r ) ≥ c ⁢ | E | 4 q {W(r)\geq\frac{c|E|^{4}}{q}} for any non-zero square number r ∈ 𝔽 q {r\in\mathbb{F}_{q}} , where C is a sufficiently large constant, c is some number between 0 and 1, and | E | {|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 ) {W(r)} to a quadratic homogeneous variety in 2 ⁢ d {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.