Some reductions of the spectral set conjecture to integers

    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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