Fuglede's Conjecture for Cyclic Groups of Square-free Order

Hu Tan; Ying Zhang

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arXiv · 2610.02218

Fuglede's Conjecture for Cyclic Groups of Square-free Order

Hu Tan·Ying Zhang

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Abstract

We prove Fuglede's conjecture for finite cyclic groups of square-free order. Specifically, if $N$ is square-free and $A\subseteq \mathbb Z_N$ is nonempty with $|A|=m$, then $A$ is spectral if and only if it tiles $\mathbb Z_N$; moreover, these conditions hold precisely when $m\mid N$ and $A$ is a complete set of representatives modulo $m$. This gives a complete structural description of spectral and tiling sets in $\mathbb Z_N$. The key new ingredient is a prime-coordinate elimination theorem: if $P\nmid nm$ is an odd prime, then every $m$-point spectral pair in $\mathbb Z_n\times\mathbb Z_P$ projects injectively to a spectral pair in $\mathbb Z_n$. Its proof combines character-matrix orthogonality, cyclotomic coefficient comparison, and a $P$-adic divisibility argument for associated Gram matrices. Iterating this reduction yields the result for arbitrary square-free $N$, with no restriction on the number or relative sizes of its prime factors.

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BibTeX RIS

Hu Tan, Ying Zhang. 2026-09-09. Fuglede's Conjecture for Cyclic Groups of Square-free Order. https://arxiv.org/abs/2610.02218

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