Final published version
Licence: CC BY: Creative Commons Attribution 4.0 International License
Research output: Contribution to Journal/Magazine › Editorial › peer-review
Article number | 14 |
---|---|
<mark>Journal publication date</mark> | 17/09/2021 |
<mark>Journal</mark> | Discrete Analysis |
Volume | 2021 |
Publication Status | Published |
<mark>Original language</mark> | English |
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.