Counting monochromatic solutions to diagonal Diophantine equations

Prendiville, Sean (2020) Counting monochromatic solutions to diagonal Diophantine equations. arXiv.

Text (2003.10161v1)
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Abstract

Given a finite colouring of the positive integers, we count monochromatic solutions to a variety of Diophantine equations, each of which can be written by setting a diagonal quadratic form equal to a linear form. As a consequence, we determine an algebraic criterion for when such equations are partition regular. Our methods involve discrete harmonic analysis and require a number of `mixed' restriction estimates, which may be of independent interest.