Title

Some Reductions Of The Spectral Set Conjecture To Integers

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 R1, there are many results in the literature that discuss the relations among various forms of the Fuglede conjecture on Zn, Z and R1 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 R1 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 [1], and we show that if the spectral sets and the tiles in Z satisfy the Coven-Meyerowitz property, then both sides of the Fuglede conjecture on R1 are true. © 2013 Cambridge Philosophical Society.

Publication Date

1-1-2014

Publication Title

Mathematical Proceedings of the Cambridge Philosophical Society

Volume

156

Issue

1

Number of Pages

123-135

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1017/S0305004113000558

Socpus ID

84890681446 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84890681446

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