On Fuglede’s conjecture and the existence of universal spectra
نویسندگان
چکیده
R 5 Tao produced a spectral set which is not a tile, while Kolountzakis and Matolcsi showed an example of a non-spectral tile. In search of lower dimensional nonspectral tiles we were led to investigate the Universal Spectrum Conjecture (USC) of Lagarias and Wang [13]. In particular, we prove here that the USC and the “tile → spectral” direction of Fuglede’s conjecture are equivalent in any dimensions. Also, we show by an example that the sufficient condition of Lagarias and Szabó [12] for the existence of universal spectra is not necessary. This fact causes considerable difficulties in producing lower dimensional examples of tiles which have no spectra. We overcome these difficulties by invoking some ideas of Révész and Farkas [2], and obtain non-spectral tiles in R. Fuglede’s conjecture and the Universal Spectrum Conjecture remains open in 1 and 2 dimensions. The 1 dimensional case is closely related to a number theoretical conjecture on tilings by Coven and Meyerowitz [1].
منابع مشابه
Spectra of certain types of polynomials and tiling of integers with translates of finite sets
Definition 1.1 is motivated by a conjecture of Fuglede [2], which asserts that a measurable set E ⊂ Rn tiles Rn by translations if and only if the space L2(E) has an orthogonal basis consisting of exponential functions {e}λ∈Λ; the set Λ is called a spectrum for E. For recent work on Fuglede’s conjecture see e.g. [4], [5], [6], [7], [8], [9], [11], [12], [13], [14], [15], [16], [18], [20], [21],...
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تاریخ انتشار 2008