Extremal Sidon sets are Fourier uniform, with applications to partition regularity

Abstract

Generalising results of Erdős-Freud and Lindström, we prove that the largest Sidon subset of a bounded interval of integers is equidistributed in Bohr neighbourhoods. We establish this by showing that extremal Sidon sets are Fourier-pseudorandom, in that they have no large non-trivial Fourier coefficients. As a further application we deduce that, for any partition regular equation in five or more variables, every finite colouring of an extremal Sidon set has a monochromatic solution. Contents1. Introduction 1 2. Fourier uniformity 6 3. Equidistribution in progressions 8 4. Equidistribution in Bohr neighbourhoods 9 5. Partition regularity 11 6. Improving Fourier uniformity 13 Appendix A. The size of a Sidon set 14 Appendix B. Trigonometric approximation 14 Appendix C. A dense model lemma 15 References 18