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PreprintCC BY 4.0arXiv

Energy Estimation of the Hamming Slice and its Applications

  • Aniruddha Biswas
  • Jihun Hwang
  • Hemanta K. Maji
  • Ilya D. Shkredov
  • Xiuyu Ye
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Abstract

Let R=ℤ/(2n−1)ℤ, where n≥ 3, and let Sw⊆ R be the residues whose canonical n-digit binary expansion has Hamming weight w. We obtain, in particular, an asymptotic formula for the additive energy of Sw E(Sw)=|Sw|4/|R|+ 𝒪(|R|3 n−3 ), which holds uniformly in w. The error term is optimal in order, with a matching lower bound for w=⌊ n/2+√n ⌋. It follows that triple sums of arbitrary unit dilates have asymptotically uniform representation counts when ∏j=13 |Swj | /(|R| n−3/5)3→∞, and that double sums have asymptotically full support when |Sw1| |Sw2 |/(|R| n−3/4)2→∞. In the proof, modular collisions are represented using a cyclic binary carry automaton; this appears to be a novel approach in this area of problems.

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Cite

BibTeXRIS
BibTeX
@misc{biswas2026energy,
  title = {{Energy Estimation of the Hamming Slice and its Applications}},
  author = {Aniruddha Biswas and Jihun Hwang and Hemanta K. Maji and Ilya D. Shkredov and Xiuyu Ye},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08056},
  eprint = {2609.08056},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08056},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Energy Estimation of the Hamming Slice and its Applications
AU  - Aniruddha Biswas
AU  - Jihun Hwang
AU  - Hemanta K. Maji
AU  - Ilya D. Shkredov
AU  - Xiuyu Ye
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.08056
UR  - https://arxiv.org/abs/2609.08056
AB  - Let R=ℤ/(2^n−1)ℤ, where n≥ 3, and let S_w⊆ R be the residues whose canonical n-digit binary expansion has Hamming weight w. We obtain, in particular, an asymptotic formula for the additive energy of S_w E(S_w)=|S_w|^4/|R|+ 𝒪(|R|^3 n^−3 ), which holds uniformly in w. The error term is optimal in order, with a matching lower bound for w=⌊ n/2+√n ⌋. It follows that triple sums of arbitrary unit dilates have asymptotically uniform representation counts when ∏_j=1^3 |S_w_j | /(|R| n^−3/5)^3→∞, and that double sums have asymptotically full support when |S_w_1| |S_w_2 |/(|R| n^−3/4)^2→∞. In the proof, modular collisions are represented using a cyclic binary carry automaton; this appears to be a novel approach in this area of problems.
ER  -