On the structure of spectral and tiling subsets of cyclic groups

Romanos Diogenes Malikiosis

Forum of Mathematics, Sigma, Tome 10 (2022)

Résumé

The purpose of this paper is to investigate the properties of spectral and tiling subsets of cyclic groups, with an eye towards the spectral set conjecture [9] in one dimension, which states that a bounded measurable subset of $\mathbb {R}$ accepts an orthogonal basis of exponentials if and only if it tiles $\mathbb {R}$ by translations. This conjecture is strongly connected to its discrete counterpart, namely that, in every finite cyclic group, a subset is spectral if and only if it is a tile. The tools presented herein are refinements of recent ones used in the setting of cyclic groups; the structure of vanishing sums of roots of unity [20] is a prevalent notion throughout the text, as well as the structure of tiling subsets of integers [1]. We manage to prove the conjecture for cyclic groups of order $p^{m}q^{n}$, when one of the exponents is $\leq 6$ or when $p^{m-2}$, and also prove that a tiling subset of a cyclic group of order $p_{1}^{m}p_{2}\dotsm p_{n}$ is spectral.

Détail

DOI : 10.1017/fms.2022.14