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

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

2009-03-10

arxiv.org/abs/0903.1834v1

Mathematics

Number Theory

24+epsilon pages

Scientific paper

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.

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