Tiling properties of spectra of measures

    Authors

    D. E. Dutkay;J. Haussermann

    Comments

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    Abbreviated Journal Title

    Math. Z.

    Keywords

    Spectrum; Tile; Hadamard matrix; Fractal; Affine iterated function; system; Cantor set; Fuglede conjecture; Coven-Meyerowitz conjecture; ITERATED FUNCTION SYSTEMS; FUGLEDES CONJECTURE; CANTOR MEASURES; MATRICES; SET; Mathematics

    Abstract

    We investigate tiling properties of spectra of measures, i.e., sets in such that forms an orthogonal basis in , where is some finite Borel measure on . Such measures include Lebesgue measure on bounded Borel subsets, finite atomic measures and some fractal Hausdorff measures. We show that various classes of such spectra of measures have translational tiling properties. This lead to some surprizing tiling properties for spectra of fractal measures, the existence of complementing sets and spectra for finite sets with the Coven-Meyerowitz property, the existence of complementing Hadamard pairs in the case of Hadamard pairs of size 2, 3, 4 or 5. In the context of the Fuglede conjecture, we prove that any spectral set is a tile, if the period of the spectrum is 2, 3, 4 or 5.

    Journal Title

    Mathematische Zeitschrift

    Volume

    280

    Issue/Number

    1-2

    Publication Date

    1-1-2015

    Document Type

    Article

    Language

    English

    First Page

    525

    Last Page

    549

    WOS Identifier

    WOS:000354242000024

    ISSN

    0025-5874

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