Unsupported Cyclotomic Divisors in Three-Prime Integer Tilings
Abstract
Cyclotomic divisibility imposes strong prime-power structure on integer tiles. We study unsupported cyclotomic divisors: mixed-order divisors for which none of the prime-power components of the order divides the mask, although every prime in the order divides the tile cardinality. Kiss, Łaba, Marshall and Somlai asked whether such a phenomenon can occur in the three-prime setting. We prove that unsupported cyclotomic divisors already occur for periods with three distinct prime factors. For primes p<q<r, we characterize the square-period case: an unsupported factor Φpqr occurs in a tiling of ZZ(pqr)2 if and only if r∈⟨ p,q⟩, and every such tile lies in a single residue class modulo r. Among cyclic tilings with the unsupported order dividing the specified modulus, the smallest modulus is 180; if the order has three distinct prime factors, it is 900. An Apéry-set construction gives examples for every triple at period p2q2r3.The proof of our results combines Fourier rigidity, a three-cylinder decomposition, and an integer mass obstruction.
Related
- Cyclotomic Newton Expansions and a Rank-Uniform Integer-Valued Newton Completion
- Three omitted values and non-Blaschke point divisors in half-planes
- External Difference Families Arising from Two or Three Cyclotomic Classes
- The average number of representations of an integer as a sum of three or more primes, over multiples of a fixed integer
- Integer decomposition property of three-dimensional smooth polytopes: a shorter proof
BibTeX
@misc{tan2026unsupported,
title = {{Unsupported Cyclotomic Divisors in Three-Prime Integer Tilings}},
author = {Hu Tan and Ying Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06677},
eprint = {2609.06677},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06677},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Unsupported Cyclotomic Divisors in Three-Prime Integer Tilings
AU - Hu Tan
AU - Ying Zhang
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06677
UR - https://arxiv.org/abs/2609.06677
AB - Cyclotomic divisibility imposes strong prime-power structure on integer tiles. We study unsupported cyclotomic divisors: mixed-order divisors for which none of the prime-power components of the order divides the mask, although every prime in the order divides the tile cardinality. Kiss, Łaba, Marshall and Somlai asked whether such a phenomenon can occur in the three-prime setting. We prove that unsupported cyclotomic divisors already occur for periods with three distinct prime factors. For primes p<q<r, we characterize the square-period case: an unsupported factor Φ_pqr occurs in a tiling of ZZ_(pqr)^2 if and only if r∈⟨ p,q⟩, and every such tile lies in a single residue class modulo r. Among cyclic tilings with the unsupported order dividing the specified modulus, the smallest modulus is 180; if the order has three distinct prime factors, it is 900. An Apéry-set construction gives examples for every triple at period p^2q^2r^3.The proof of our results combines Fourier rigidity, a three-cylinder decomposition, and an integer mass obstruction.
ER -