Title

Some reductions of the spectral set conjecture to integers

Authors

Authors

D. E. Dutkay;C. K. Lai

Comments

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Abbreviated Journal Title

Math. Proc. Camb. Philos. Soc.

Keywords

FINITE ABELIAN-GROUPS; FUGLEDES CONJECTURE; UNIVERSAL SPECTRA; FACTORIZATION; LINE; Mathematics

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 R-1, there are many results in the literature that discuss the relations among various forms of the Fuglede conjecture on Z(n), Z and 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 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 [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 R-1 are true.

Journal Title

Mathematical Proceedings of the Cambridge Philosophical Society

Volume

156

Issue/Number

1

Publication Date

1-1-2014

Document Type

Article

Language

English

First Page

123

Last Page

135

WOS Identifier

WOS:000337084300007

ISSN

0305-0041

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