Some reductions of the spectral set conjecture to integers

TL;DR

This paper explores the connections between the spectral set conjecture and tiling in one dimension, reducing the problem to integer cases and examining properties like the Coven-Meyerowitz condition.

Contribution

It establishes equivalences among various forms of the conjecture in dimension one and links the conjecture to the Coven-Meyerowitz property for integers.

Findings

Fuglede conjecture equivalences in dimension one clarified

Spectral sets with rational measure have rational spectra if conjecture holds

Coven-Meyerowitz property implies the conjecture in one dimension

Abstract

The spectral set conjecture, also known as the Fuglede conjecture, asserts that every bounded spectral set is a tile and vice versa. While this conjecture remains open on ${\mathbb R}^1$, there are many results in the literature that discuss the relations among various forms of the Fuglede conjecture on ${\mathbb Z}_n$, ${\mathbb Z}$ and ${\mathbb R}^1$ and also the seemingly stronger universal tiling (spectrum) conjectures on the respective groups. In this paper, we clarify the equivalences between these statements in dimension one. In addition, we show that if the Fuglede conjecture on ${\mathbb R}^1$ is true, then every spectral set with rational measure must have a rational spectrum. We then investigate the Coven-Meyerowitz property for finite sets of integers, introduced in \cite{CoMe99}, and we show that if the spectral sets and the tiles in ${\mathbb Z}$ satisfy the Coven-Meyerowitz property, then both sides of the Fuglede conjecture on ${\mathbb R}^1$ are true.