Some new thin sets of integers in Harmonic Analysis

TL;DR

This paper constructs new types of integer sets in Harmonic Analysis that are simultaneously small in some harmonic senses and large in others, challenging previous notions of thin sets.

Contribution

It introduces a probabilistic method to build integer sets with both smallness and largeness properties, contrasting with known thin sets.

Findings

Sets are close to Sidon sets with uniformly convergent Fourier series.

Fourier coefficients of functions with spectrum in these sets are in all  spaces for p>1.

The space of bounded functions with spectrum in these sets is non-separable.

Abstract

We randomly construct various subsets $\Lambda$ of the integers which have both smallness and largeness properties. They are small since they are very close, in various meanings, to Sidon sets: the continuous functions with spectrum in $\Lambda$ have uniformly convergent series, and their Fourier coefficients are in $\ell_p$ for all $p>1$; moreover, all the Lebesgue spaces $L^q_\Lambda$ are equal for $q<+\infty$. On the other hand, they are large in the sense that they are dense in the Bohr group and that the space of the bounded functions with spectrum in $\Lambda$ is non separable. So these sets are very different from the thin sets of integers previously known.