On the structure of spectral and tiling subsets of cyclic groups

Abstract

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 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 is a prevalent notion throughout the text, as well as the structure of tiling subsets of integers. We manage to prove the conjecture for cyclic groups of order $p^m{q}^n$, when one of the exponents is $\le 6$ or when $p^{m-2}<q^4$, and also prove that a tiling subset of a cyclic group of order $p_1^m{p}_2\cdots p_n$ is spectral.

38 pages; fixed typos, incorporated referee's suggestions, added one section outlining the technical aspects of the proof of Theorem 1.4

References in corpus (5)

• Complex Hadamard matrices and the Spectral Set Conjecture
• The Fuglede conjecture for convex domains is true in all dimensions
• On Fuglede's conjecture and the existence of universal spectra
• Fuglede's conjecture fails in 4 dimensions over odd prime fields
• Equi-distributed property and spectral set conjecture on ${\mathbb{Z}}_{p^2}\times {\mathbb{Z}}_p$

Cited by in corpus (4)

• Fuglede's conjecture holds for cyclic groups of order $pqrs$
• Spectral sets and tiles in ${\mathbb{Z}}_p^2\times {\mathbb{Z}}_q^2$
• Fuglede's conjecture holds in ${\mathbb{Z}}_p\times {\mathbb{Z}}_{p^n}$
• Polynomial method for perfect 2-colourings of circulant graphs