Counting monochromatic solutions to diagonal Diophantine equations

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.

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Cited by in corpus (3)

• On monochromatic solutions to $x-y=z^2$
• Arithmetic Ramsey theory over the primes
• Partition regularity of polynomial systems near zero