Counting Monochromatic Solutions to Diagonal Diophantine Equations

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

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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.

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Journal Article
Journal or Publication Title:
Discrete Analysis
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Deposited On:
09 Dec 2022 12:25
Last Modified:
09 Dec 2022 12:25