Erdos-Turan with a moving target, equidistribution of roots of reducible quadratics, and Diophantine quadruples

TL;DR

This paper develops new tools to analyze the distribution of solutions to polynomial congruences and applies them to count Diophantine quadruples, providing asymptotic formulas and extending classical inequalities.

Contribution

It extends the Erdős-Turán inequality to variable intervals and adapts Hooley's method for reducible quadratics, advancing the study of Diophantine quadruples.

Findings

Derived an asymptotic count of Diophantine quadruples up to x

Extended the Erdős-Turán inequality for variable intervals

Adapted Hooley's argument for reducible quadratic polynomials

Abstract

A Diophantine $m$-tuple is a set $A$ of $m$ positive integers such that $ab+1$ is a perfect square for every pair $a,b$ of distinct elements of $A$. We derive an asymptotic formula for the number of Diophantine quadruples whose elements are bounded by $x$. In doing so, we extend two existing tools in ways that might be of independent interest. The Erd\H os-Tur\'an inequality bounds the discrepancy between the number of elements of a sequence that lie in a particular interval modulo 1 and the expected number; we establish a version of this inequality where the interval is allowed to vary. We also adapt an argument of Hooley on the equidistribution of solutions of polynomial congruences to handle reducible quadratic polynomials.