A Proof of Fraenkel's Conjecture

@misc{tan2026proof,
  title = {{A Proof of Fraenkel's Conjecture}},
  author = {Hu Tan and Ying Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.01570},
  eprint = {2609.01570},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01570},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - A Proof of Fraenkel's Conjecture
AU  - Hu Tan
AU  - Ying Zhang
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.01570
UR  - https://arxiv.org/abs/2609.01570
AB  - Fraenkel's conjecture asserts that a partition of the integers into at least three Beatty sequences with distinct moduli has the binary densities 1,2,4,…,2^m−1, normalized by 2^m−1. We prove the conjecture through a dimension-free intermediate statement: every such partition contains a component of density at least 1/3. After reducing the partition to primitive common-period data, Fourier cancellation produces a finite inverse-sine system. We prove that no such system can exist when every density is below 1/3. The proof combines a divisor-concentration identity with uniform analytic estimates and three exact finite verifications, all carried out with integer or rational arithmetic. The component supplied by the density bound has mean spacing at most three. Deleting it preserves balance, and every surviving periodic balanced set is again a rational Beatty set. Induction determines the surviving binary scales, while a two-sequence disjointness criterion forces the deleted density to be the next binary scale. This yields the asserted density pattern.An eventual Beatty representation of balanced binary indicators extends the classification to one-sided balanced sequences on at least three letters with distinct positive densities, proving the balanced-sequence conjecture of Altman, Gaujal, and Hordijk.
ER  -