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

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-Turán 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.

arXiv comment24+epsilon pages

Cited by in corpus (4)

• Asymptotics of $D(q)$-pairs and triples via $L$-functions of Dirichlet charaters
• On the largest element in D(n)-quadruples
• Exponential sums with reducible polynomials
• There is no Diophantine quintuple