Counting Monochromatic Solutions to Diagonal Diophantine Equations

Prendiville, Sean (2021) Counting Monochromatic Solutions to Diagonal Diophantine Equations. Discrete Analysis, 2021: 14. ISSN 2397-3129

Abstract

We show how to adapt the Hardy–Littlewood circle method to count monochromatic solutions to diagonal Diophantine equations. This delivers a lower bound which is optimal up to absolute constants. The method is illustrated on equations obtained 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.