Abstract
Let A be a finite set of integers. We say that A tiles the integers if there is a set T ⊆ ℤ such that {t+A: t∈T{ forms a disjoint partition of the integers. It has long been known that such a set T must be periodic. The question is to determine how long the period of T can become as a function of the diameter of the set A. The previous best lower bound, due to Kolountzakis [7], shows that the period of T can grow as fast as the square of the diameter of A. In this paper we improve Kolountzakis’ lower bound by showing that the period of T can in fact grow faster than any power of the diameter of A.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
A. Bíró: Divisibility of integer polynomials and tilings of the integers, Acta Arithmetica118 (2005), 117–127.
N. G. de Bruijn: On the factorization of cyclic groups, Indag. Math. 15 (1953), 370–377.
E. M. Coven and A. Meyerowitz: Tiling the integers with translates of one finite set, J. Algebra212(1) (1999), 161–174.
A. Granville, I. Laba and Y. Wang: On finite sets which tile the integers, preprint (2001).
Gy. Hajós: Sur la factorisation des groupes abéliens, Časopis Pěest. Mat. Fys.74 (1950), 157–162.
Gy. Hajós: Sur la problème de factorisation des groupes cycliques, Acta. Math. Sci. Hungar. 1 (1950), 189–195.
M. N. Kolountzakis: Translational tilings of the integers with long periods, Electronic Journal of Combinatorics10(1) (2003), #R22.
M. N. Kolountzakis and M. Matolcsi: Tiles with no spectra, Forum Math. 18(3) (2006), 519–528.
C. T. Long: Addition theorems for sets of integers, Pacific J. Math. 23(1) (1967), 107–112.
D. J. Newman: Tesselation of integers, J. Number Theory9 (1977), 107–111.
L. Rédei: Über das Kreisteilungspolynom, Acta Math. Acad. Sci. Hungar. 5 (1954), 27–28.
I. Z. Ruzsa: Appendix, in: R. Tijdeman: Periodicity and Almost-periodicity, preprint (August 2002).
A. D. Sands: On a conjecture of G. Hajós, Glasgow Math. Journal15 (1974), 88–89.
S. K. Stein and S. Szabó: Algebra and Tilings, Mathematical Association of America Carus Mathematical Monograph series No. 25 (1994).
J. P. Steinberger: A class of prototiles with doubly generated level semigroups, Journal of Combinatorial Theory, Series A106(1) (2004), 1–13.
J. P. Steinberger: Multiple tilings of ℤ with long periods and tiles with manygenerated level semigroups, New York Journal of Mathematics11 (2005), 445–456.
J. P. Steinberger: Indecomposable tilings of the integers with exponentially long periods, Electronic Journal of Combinatorics12 (2005), #R36.
J. P. Steinberger: Quasiperiodic group factorizations, Resultate der Mathematik51(3–4) (2008), 319–338.
C. Swenson: Direct sum subset decompositions of ℤ, Pacific J. Math.53 (1974), 629–633.
S. Szabó: Quasi-periodic factorizations, personal communication.
S. Szabó: Topics in factorization of Abelian groups, Birkhauser (2004).
R. Tijdeman: Decomposition of the integers as a direct sum of two subsets, in: Number Theory, Paris, 1992–1993.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Steinberger, J.P. Tilings of the integers can have superpolynomial periods. Combinatorica 29, 503–509 (2009). https://doi.org/10.1007/s00493-009-2269-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-009-2269-9