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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.07135 |
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| _version_ | 1866908855098671104 |
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| author | Dai, Xin-Rong |
| author_facet | Dai, Xin-Rong |
| contents | In the study of factorizations of finite cyclic groups, a classical problem is to investigate the properties of factorization sets $A$ and $B$ in the direct sum decomposition $A \oplus B = \mathbb{Z}_{M}$ with $|A| = |B| =\sqrt{M}$, where $M=(pqr)^2$ for some distinct primes $p$, $q$, and $r$. In this paper, we show that neither $A$ nor $B$ is contained in a proper subgroup of $\mathbb{Z}_{(pqr)^2}$ if and only if the factorization sets $A, B$ form a Szabó pair. The factorization of finite cyclic groups is closely connected to the properties of tiling and spectral sets in $\Bbb Z$. The problem considered in this paper is equivalent to the simplest form of tiling that cannot be reduced to the two--prime case by the method provided by Coven and Meyerowitz (J. Algebra 212: 161--174, 1999). In contrast, the construction for the tiling which can be reduced to the two--prime case is already known. Our results present full structures for the factorization sets $A$ and $B$, and therefore, for this class of tilings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_07135 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Factorization of Finite Cyclic Group $\Bbb Z_{(pqr)^2}$: Szabó Pairs and Full Tiling Structures Dai, Xin-Rong Combinatorics 20K25, 05B45, 11B75, 11C08, 52C22 In the study of factorizations of finite cyclic groups, a classical problem is to investigate the properties of factorization sets $A$ and $B$ in the direct sum decomposition $A \oplus B = \mathbb{Z}_{M}$ with $|A| = |B| =\sqrt{M}$, where $M=(pqr)^2$ for some distinct primes $p$, $q$, and $r$. In this paper, we show that neither $A$ nor $B$ is contained in a proper subgroup of $\mathbb{Z}_{(pqr)^2}$ if and only if the factorization sets $A, B$ form a Szabó pair. The factorization of finite cyclic groups is closely connected to the properties of tiling and spectral sets in $\Bbb Z$. The problem considered in this paper is equivalent to the simplest form of tiling that cannot be reduced to the two--prime case by the method provided by Coven and Meyerowitz (J. Algebra 212: 161--174, 1999). In contrast, the construction for the tiling which can be reduced to the two--prime case is already known. Our results present full structures for the factorization sets $A$ and $B$, and therefore, for this class of tilings. |
| title | Factorization of Finite Cyclic Group $\Bbb Z_{(pqr)^2}$: Szabó Pairs and Full Tiling Structures |
| topic | Combinatorics 20K25, 05B45, 11B75, 11C08, 52C22 |
| url | https://arxiv.org/abs/2601.07135 |