Result: On the minimal period of integer tilings

Title:
On the minimal period of integer tilings
Authors:
Izabella Łaba, Dmitrii Zakharov
Source:
Bulletin of the London Mathematical Society. 57:1160-1170
Publication Status:
Preprint
Publisher Information:
Wiley, 2025.
Publication Year:
2025
Subject Terms:
Mathematics - Number Theory, 05B45, 52C22, Coven-Meyerowitz conjecture, Lattice packing and covering (number-theoretic aspects), factorization, Combinatorial aspects of tessellation and tiling problems, Other combinatorial number theory, Special sequences and polynomials, tiling, FOS: Mathematics, Mathematics - Combinatorics, Number Theory (math.NT), Combinatorics (math.CO), minimal tiling period estimates
Document Type:
Academic journal Article
File Description:
application/xml
Language:
English
ISSN:
1469-2120
0024-6093
DOI:
10.1112/blms.70023
DOI:
10.48550/arxiv.2406.14824
Access URL:
http://arxiv.org/abs/2406.14824
Rights:
CC BY NC
arXiv Non-Exclusive Distribution
Accession Number:
edsair.doi.dedup.....37e46116905991d33c784537879b7258
Database:
OpenAIRE

Further Information

If a finite set tiles the integers by translations, it also admits a tiling whose period has the same prime factors as . We prove that the minimal period of such a tiling is bounded by , where is the diameter of . In the converse direction, given , we construct tilings whose minimal period has the same prime factors as and is bounded from below by . We also discuss the relationship between minimal tiling period estimates and the Coven–Meyerowitz conjecture.