Decompositions for Cyclic Groups with 3 Prime Factors

TL;DR

This work characterizes factorizations $A\oplus B=\mathbb{Z}_{(pqr)^2}$ with $|A|=|B|=pqr$ (and $0\in A\cap B$) for distinct primes $p,q,r$, showing that such a factorization is possible if and only if the pair is Szabó-type (up to translation). The authors build a framework based on division sets and cyclotomic divisors, establishing that $\Phi_p(x)\Phi_q(x)\Phi_r(x)$ divides $A(x)$ and that $p,q,r$ lie in $\mathrm{Div}(B)$ while their squares do not. They then derive a detailed, translate-invariant description of $B$ (the Szabó-type decomposition) and prove that the only viable factorizations arise from Szabó pairs, thereby connecting Sands’ criterion with the Szabó construction in the $(pqr)^2$ setting. The results advance the understanding of 1-dimensional spectral tiling and illuminate the structural parallels with Coven–Meyerowitz’s CM framework for finite cyclic groups.

Abstract

In this paper, we characterize the direct sum decompositions of the cyclic group $\mathbb{Z}_{(pqr)^2}$, where $p$, $q$, and $r$ are distinct primes. We show that if $A \oplus B = \mathbb{Z}_{(pqr)^2}$ with $|A| = |B| = pqr$, then Sands' conjecture fails to hold, in other words, neither $A$ nor $B$ is contained in a proper subgroup of $\mathbb{Z}_{(pqr)^2}$, if and only if the sets $A, B$ form a Szabó pair.

Theorems & Definitions (20)

• Definition 1.1
• Theorem 1.1
• Proposition 2.1
• Theorem 2.2
• Proposition 2.3
• Lemma 2.4
• Proposition 2.5
• Corollary 2.6
• Proposition 2.7
• Corollary 2.8