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ERDŐS–TURÁN WITH A MOVING TARGET, EQUIDISTRIBUTION OF ROOTS OF REDUCIBLE QUADRATICS, AND DIOPHANTINE QUADRUPLES

Part of: Multiplicative number theory Diophantine equations Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  13 December 2010

Greg Martin  and
Scott Sitar
Greg Martin
Affiliation:
Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Canada V6T 1Z2 (email: gerg@math.ubc.ca)
Scott Sitar
Affiliation:
Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Canada V6T 1Z2 (email: sesitar@math.ubc.ca)
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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 Erdős–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.

MSC classification

Secondary: 11D45: Counting solutions of Diophantine equations 11N45: Asymptotic results on counting functions for algebraic and topological structures 11K06: General theory of distribution modulo $1$ 11K38: Irregularities of distribution, discrepancy
Type
Research Article
Information
Mathematika , Volume 57 , Issue 1 , January 2011 , pp. 1 - 29
Copyright
Copyright © University College London 2011

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References

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