Universal Spectra , Universal Tiling Sets and Thespectral Set
نویسنده
چکیده
A subset of R d with nite positive Lebesgue measure is called a spectral set if there exists a subset R such that E := n e i2h;xi : 2 o form an orthogonal basis of L 2 ((). The set is called a spectrum of the set. The Spectral Set Conjecture states that is a spectral set if and only if tiles R d by translation. In this paper we prove the Spectral Set Conjecture for a class of sets R. Speciically we show that a spectral set possessing a spectrum that is a strongly periodic set must tile R by translates of a strongly periodic set depending only on the spectrum, and vice versa.
منابع مشابه
Universal Spectra and Tijdeman’s Conjecture on Factorization of Cyclic Groups
ABSTRACT. A spectral set Ω in R is a set of finite Lebesgue measure such that L(Ω) has an orthogonal basis of exponentials {e2πi〈λ,x〉 : λ ∈ Λ} restricted to Ω. Any such set Λ is called a spectrum for Ω. It is conjectured that every spectral set Ω tiles R by translations. A tiling set T of translations has a universal spectrum Λ if every set Ω that tiles R by T is a spectral set with spectrum Λ....
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