Salem sets, equidistribution and arithmetic progressions

Abstract

Arithmetic progressions of length \(3\) may be found in compact subsets of the reals that satisfy certain Fourier-dimensional as well as Hausdorff-dimensional requirements. It has been shown that a very similar result holds in the integers under analogous conditions, with Fourier dimension being replaced by the decay of a discrete Fourier transform. By using a construction of Salem's, we show that this correspondence can be made more precise. Specifically, we show that a subset of the integers can be mapped to a compact subset of the continuum in a way which preserves equidistribution properties as well as arithmetic progressions of arbitrary length, and vice versa. We use the method to characterise Salem sets in \(\mathbb{R}\) through discrete, equidistributed approximations. Finally, we discuss how this method sheds light on the generation of Salem sets by stationary stochastic processes.