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

2002/09/16 by Sergeĭ Konyagin, Sergei Konyagin, Konyagin, Sergei +3
Mathematics · #11A99 #FOS: Mathematics #Mathematical Analysis and Transform Methods #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Spectral Theory in Mathematical Physics #math.NT #msc:11A99

paper · pdf · doi:10.48550/arxiv.math/0209204

14 pages

arxiv created 2002/09/16 · openalex publication_date 2002/09/16 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider two number-theoretic problems arising from Fuglede's spectral set conjecture: characterizing finite sets that tile integers, and finding polynomials with (0,1) coefficients whose roots have a certain multiplicative structure. We verify several special cases of the relevant conjectures; in particular, we find necessary and sufficient conditions for a set A to tile the integers if A is a direct sum of cyclic subsets.

Related

• On spectral sets of integers
• Tiling sets and spectral sets over finite fields
• A group ring approach to Fuglede's conjecture in cyclic groups
• On the universal tiling conjecture in dimension one
• Tiling the integers with translates of one finite set