Distribution of integer points on determinant surfaces and a $\text{mod-}p$ analogue

TL;DR

This paper derives asymptotic formulas for counting integer solutions to the equation xy - zw = r and the congruence xy - zw ≡ 1 (mod p), providing explicit main terms and strong error bounds.

Contribution

It introduces explicit asymptotic formulas with error bounds for solutions to determinant surface equations and their mod p analogues.

Findings

Asymptotic formula for solutions to xy - zw = r with explicit main term.

Asymptotic formula for solutions to xy - zw ≡ 1 (mod p) with error bounds.

Strong bounds on error terms in both cases.

Abstract

We establish an asymptotic formula for counting integer solutions with smooth weights to an equation of the form $xy-zw=r$, where $r$ is a non-zero integer, with an explicit main term and a strong bound on the error term in terms of the size of the variables $x, y, z, w$ as well as of $r$. We also establish an asymptotic formula for counting integer solutions with smooth weights to the congruence $xy-zw \equiv 1 (\text{mod }p)$, where $p$ is a large prime, with a strong bound on the error term.