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Spectra of certain types of polynomials and tiling of integers with translates of finite sets

✍ Scribed by Sergei Konyagin; Izabella Łaba

Publisher
Elsevier Science
Year
2003
Tongue
English
Weight
259 KB
Volume
103
Category
Article
ISSN
0022-314X

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✦ Synopsis

We investigate Fuglede's spectral set conjecture in the special case when the set in question is a union of finitely many unit intervals in dimension 1. In this case, the conjecture can be reformulated as a statement about multiplicative properties of roots of associated with the set polynomials with ð0; 1Þ coefficients. Let AðxÞAz½x be an N-term polynomial. We say that fy 1 ; y 2 ; y; y NÀ1 g is an N-spectrum for AðxÞ if the y j are all distinct and A e 2piðyj Ày k Þ ¼ 0 for all 0pj; k; pN À 1; jak:

We establish necessary and sufficient conditions for irreducible polynomials and for products of two factors of the form

to have a spectrum. This confirms Fuglede's conjecture for associated sets.

📜 SIMILAR VOLUMES

Tiling the Integers with Translates of O
Tiling the Integers with Translates of One Finite Set
✍ Ethan M. Coven; Aaron Meyerowitz 📂 Article 📅 1999 🏛 Elsevier Science 🌐 English ⚖ 115 KB

A set tiles the integers if and only if the integers can be written as a disjoint union of translates of that set. We consider the problem of finding necessary and sufficient conditions for a finite set to tile the integers. For sets of prime power Ž . size, it was solved by D. Newman 1977, J. Numbe