{
  "abstract": "Let $R=\\mathbb{Z}/(2^n-1)\\mathbb{Z}$, where $n\\geq 3$, and let $S_w\\subseteq R$ 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)=\\frac{\\left|S_w\\right|^4}{|R|}+ \\mathcal{O}\\left(|R|^3 n^{-3} \\right), \\] which holds uniformly in $w$. The error term is optimal in order, with a matching lower bound for $w=\\lfloor n/2+\\sqrt{n} \\rfloor$. It follows that triple sums of arbitrary unit dilates have asymptotically uniform representation counts when $\\prod_{j=1}^{3} \\left|S_{w_j} \\right| /\\left(|R| n^{-3/5}\\right)^3\\to\\infty$, and that double sums have asymptotically full support when $\\left|S_{w_1}\\right| \\left|S_{w_2} \\right|/\\left(|R| n^{-3/4}\\right)^2\\to\\infty$. 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08056",
  "author_total": 5,
  "authors": [
    {
      "name": "Aniruddha Biswas"
    },
    {
      "name": "Jihun Hwang"
    },
    {
      "name": "Hemanta K. Maji"
    },
    {
      "name": "Ilya D. Shkredov"
    },
    {
      "name": "Xiuyu Ye"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1t4vb2kcfsqbq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08056",
  "fulltext_url": null,
  "id": "1t4vb2kcfsqbq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08056",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7212022414",
  "pdf_url": "https://arxiv.org/pdf/2609.08056",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1t4vb2kcfsqbq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08056",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08056"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Energy Estimation of the Hamming Slice and its Applications",
  "title_text": "Energy Estimation of the Hamming Slice and its Applications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/1t4vb2kcfsqbq",
  "venue": "arXiv",
  "versions": []
}