Equidistribution of Diophantine pairs among the equivalence classes of quadratic forms

TL;DR

The paper analyzes the distribution of Diophantine pairs with $ac+n$ a square by mapping each pair to an integral binary quadratic form of discriminant $4n$ and studying their $SL_2(\mathbb{Z})$-equivalence classes. Using a homogeneous-dynamics approach (involving $G=\mathrm{SL}_2(\mathbb{R})$ acting on binary forms and Eskin–McMullen counting, as well as Oh–Shah methods in the split case), the authors prove that $D(n)$-pairs are asymptotically equidistributed among proper equivalence classes with fixed content. This yields a streamlined proof of Badesa's asymptotic for the total number of $D(n)$-pairs by summing class-by-class counts and applying ring-class-number formulas. The results split into three regimes (definite, indefinite nonsplit, and split), delivering explicit leading-term constants tied to class numbers, fundamental units, and regulators, and illuminating the connection between Diophantine tuples and the arithmetic of binary quadratic forms.

Abstract

For a fixed integer n, a pair of nonzero integers {a, c} is called a D(n)-pair if the product ac plus n is a perfect square. In this short note we prove that D(n)-pairs are asymptotically equidistributed (via their associated quadratic forms) among proper SL_2(Z)-equivalence classes of binary quadratic forms of discriminant 4n with fixed content. As a consequence, we obtain a more streamlined and simpler proof of Badesa's asymptotic formula for the number of D(n)-pairs.

Theorems & Definitions (19)

• Theorem 2.1
• Theorem 2.2
• Theorem 3.1
• Lemma 3.2
• proof
• proof : Proof of Theorem \ref{['thm:total-count']}
• Theorem 4.1
• Lemma 5.1
• proof
• Lemma 5.2