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

@misc{tan2026fugledes,
  title = {{Fuglede's Conjecture for Cyclic Groups of Square-free Order}},
  author = {Hu Tan and Ying Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02218},
  eprint = {2610.02218},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02218},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Fuglede's Conjecture for Cyclic Groups of Square-free Order
AU  - Hu Tan
AU  - Ying Zhang
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.02218
UR  - https://arxiv.org/abs/2610.02218
AB  - We prove Fuglede's conjecture for finite cyclic groups of square-free order. Specifically, if N is square-free and A⊆ ℤ_N is nonempty with |A|=m, then A is spectral if and only if it tiles ℤ_N; moreover, these conditions hold precisely when m| N and A is a complete set of representatives modulo m. This gives a complete structural description of spectral and tiling sets in ℤ_N. The key new ingredient is a prime-coordinate elimination theorem: if Pnmid nm is an odd prime, then every m-point spectral pair in ℤ_n×ℤ_P projects injectively to a spectral pair in 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.
ER  -