Partition regular linear equations over Sidon sets

In this article, we show that the Sidon subsets of $[N{]}^d$ are Fourier uniform, and we prove a dense model lemma for Sidon sets. Using these results and a higher-dimensional version of Rado’s theorem, we prove that given any partition regular linear equation in $s\ge 5$ variables with nonzero coefficients, and any finite partition of a sufficiently dense Sidon subset $S$ of $[N{]}^d$, the number of monochromatic solutions to this equation is $\gg \mathrm{\mid }S{\mathrm{\mid }}^s{N}^{-d}$ for all large $N$. As a corollary of the Fourier uniformity result, we show that dense Sidon subsets of $[N{]}^d$ are equidistributed in certain arithmetic and Bohr structures. Our proofs are motivated by the arguments of Ortega and Prendiville.