Equality Cases in the Tu-Deng Conjecture Bound

Abstract

The Tu-Deng Conjecture, made in 2011, asserts that for k≥2 and 1≤ t<2^k−1, the set S_t,k={(a,b)∈{0,1,…,2^k−2}²: a+b≡ tpmod2^k−1, wt(a)+wt(b)<k} satisfies |S_t,k|≤ 2^k−1. Proofs of the conjecture were given independently by the author and by Liu, Luo and Xie in arXiv in 2026. Let r=wt(t), and let z=k−r be the number of zeros in the k-bit expansion of t. We prove that equality holds in the Tu-Deng Conjecture if and only if every cyclic distance between consecutive zeros is at least z. We give several other simple necessary and sufficient conditions, including a forbidden-pattern condition, a packing condition for the zero positions, a composition condition for the gaps, and equivalent conditions involving the coefficients of the polynomial H_W(u,v) in the proof of the Tu-Deng conjecture. We also count the equality words having a prescribed number of zeros.

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BibTeX RIS

BibTeX

@misc{cusick2026equality,
  title = {{Equality Cases in the Tu-Deng Conjecture Bound}},
  author = {Thomas W. Cusick},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12630},
  eprint = {2610.12630},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12630},
  note = {CC BY-NC-SA 4.0},
}

RIS

TY  - UNPB
TI  - Equality Cases in the Tu-Deng Conjecture Bound
AU  - Thomas W. Cusick
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12630
UR  - https://arxiv.org/abs/2610.12630
AB  - The Tu-Deng Conjecture, made in 2011, asserts that for k≥2 and 1≤ t<2^k−1, the set S_t,k={(a,b)∈{0,1,…,2^k−2}^2: a+b≡ tpmod2^k−1, wt(a)+wt(b)<k} satisfies |S_t,k|≤ 2^k−1. Proofs of the conjecture were given independently by the author and by Liu, Luo and Xie in arXiv in 2026. Let r=wt(t), and let z=k−r be the number of zeros in the k-bit expansion of t. We prove that equality holds in the Tu-Deng Conjecture if and only if every cyclic distance between consecutive zeros is at least z. We give several other simple necessary and sufficient conditions, including a forbidden-pattern condition, a packing condition for the zero positions, a composition condition for the gaps, and equivalent conditions involving the coefficients of the polynomial H_W(u,v) in the proof of the Tu-Deng conjecture. We also count the equality words having a prescribed number of zeros.
ER  -

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