Counting monochromatic solutions to diagonal Diophantine equations
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Publication:6337226
DOI10.19086/DA.28173arXiv2003.10161MaRDI QIDQ6337226
Publication date: 23 March 2020
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.
Applications of the Hardy-Littlewood method (11P55) Counting solutions of Diophantine equations (11D45) Arithmetic combinatorics; higher degree uniformity (11B30)
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