Counting solutions to the quadratic determinant equation

TL;DR

This paper derives an asymptotic formula for the number of integer solutions to a quadratic determinant equation within a bounded range, using combinatorial, analytic, and number-theoretic methods.

Contribution

It provides a new asymptotic count for solutions to the quadratic determinant equation, especially when h is close to N^2, confirming a prior conjecture in a broad setting.

Findings

Established an asymptotic formula for solutions with h near N^2.

Achieved square-root cancellation error terms in the count.

Confirmed a conjecture by Dhanda-Haynes-Prasala in a general case.

Abstract

Given $h, N \in \mathbb{N}$ satisfying $1 \leqslant h \leqslant N^2$, we prove an asymptotic formula for the number of solutions to the equation $x_1 x_2 - x_3 x_4 = h$ with $x_1, \ldots, x_4 \in [-N,N] \cap \mathbb{Z}$. We use a combination of combinatorial and analytic arguments in physical space along with bounds for Kloosterman sums. Our main result concerns the case when $h = N^2 + O(N)$, wherein we obtain square-root cancellation error terms by bypassing Kloosterman sum bounds and exploiting an additional symmetry available in this setting via Ramanujan sums. This confirms a speculation of Dhanda-Haynes-Prasala in a very general form.