Tiling Properties Of Spectra Of Measures

Keywords

Affine iterated function system; Cantor set; Coven–Meyerowitz conjecture; Fractal; Fuglede conjecture; Hadamard matrix; Spectrum; Tile

Abstract

We investigate tiling properties of spectra of measures, i.e., sets (Formula presented.) forms an orthogonal basis in (Formula presented.), where μ is some finite Borel measure on (Formula presented.). 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.

Publication Date

6-1-2015

Publication Title

Mathematische Zeitschrift

Volume

280

Issue

1-2

Number of Pages

525-549

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1007/s00209-015-1435-6

Socpus ID

84940003067 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84940003067

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