Tiling properties of spectra of measures

TL;DR

This paper explores the tiling properties of spectra of measures, revealing surprising tiling phenomena for fractal measures, and establishing conditions under which spectral sets also tile the real line, especially for small periods.

Contribution

It demonstrates that various spectra of measures exhibit translational tiling properties and connects these to the Fuglede conjecture for small periods.

Findings

Spectra of measures can have translational tiling properties.

Spectral sets with small periods (2-5) are proven to be tiles.

Existence of complementing sets and spectra for finite sets with Coven-Meyerowitz property.

Abstract

We investigate tiling properties of spectra of measures, i.e., sets $\Lambda$ in $\br$ such that $\{e^{2\pi i \lambda x}: \lambda\in\Lambda\}$ forms an orthogonal basis in $L^2(\mu)$, where $\mu$ is some finite Borel measure on $\br$. Such measures include Lebesgue measure on bounded Borel subsets, finite atomic measures and some fractal Hausdorff measures. We show that various classes of such spectra of measures have translational tiling properties. This lead to some surprizing tiling properties for spectra of fractal measures, the existence of complementing sets and spectra for finite sets with the Coven-Meyerowitz property, the existence of complementing Hadamard pairs in the case of Hadamard pairs of size 2,3,4 or 5. In the context of the Fuglede conjecture, we prove that any spectral set is a tile, if the period of the spectrum is 2,3,4 or 5.