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

TL;DR

This paper proves that extremal Sidon sets are Fourier-uniform and equidistributed, leading to applications in partition regularity of equations, by demonstrating their Fourier pseudorandomness and uniform distribution properties.

Contribution

It establishes that extremal Sidon sets are Fourier-pseudorandom and equidistributed, extending previous results and applying these properties to partition regularity of equations.

Findings

Extremal Sidon sets are Fourier-pseudorandom.

Largest Sidon subsets are equidistributed in Bohr neighborhoods.

Every finite coloring of extremal Sidon sets yields monochromatic solutions to certain equations.

Abstract

Generalising results of Erd\H{o}s-Freud and Lindstr\"om, 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.