{
  "abstract": "Let $q_m=2m/(m+1)$ and put \\[ β_0=\\frac{3}{2}+\\frac{1}{\\log 2}=2.9426950408\\ldots, \\] where $\\log$ is the natural logarithm. We give a proof scheme showing that, for every $\\varepsilon>0$, there is $C_\\varepsilon<\\infty$ such that every complex-valued function $f:\\{-1,1\\}^n\\to\\C$ of Fourier degree at most $m$ satisfies \\[ \\left(\\sum_{A\\subseteq[n]}\\abs{\\wh f(A)}^{q_m}\\right)^{1/q_m} \\le C_\\varepsilon m^{β_0+\\varepsilon}\\norm{f}_\\infty. \\] The improvement over the $m^9$ estimate of the earlier draft has two ingredients. The first three Fourier levels are estimated at the scales $m$, $m^{3/2}$, and $m^{5/3}$. These losses are encoded in the weight $M^{μ_r/r}r^B\\min\\{r,L_M\\}^8$, $L_M\\asymp_B \\log (M+1)$, with $μ_r=\\lceil3r/2\\rceil$ for $r\\ge2$. A parity-compatible central window handles $r<L_M$, while a shrinking balanced window handles $r\\ge L_M\\asymp_B\\log(M+1)$, no parity is used in this regime. The leading high-degree loss--gain factor is $\\ee\\,2^{-B}$; careful uniform bounds close the bootstrap for every $B>1/\\log2$. The formula for $μ_r$ is explained below.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21144",
  "author_total": 3,
  "authors": [
    {
      "name": "Joseph Slote"
    },
    {
      "name": "Chun-Kai Tseng"
    },
    {
      "name": "Alexander Volberg"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a385r728agm9t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21144",
  "fulltext_url": null,
  "id": "a385r728agm9t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21144",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7213881609",
  "pdf_url": "https://arxiv.org/pdf/2609.21144",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a385r728agm9t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21144",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21144"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "An $m^{2.943}$ Bohnenblust--Hille Bound on the Boolean Cube",
  "title_text": "An m^2.943 Bohnenblust--Hille Bound on the Boolean Cube",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/a385r728agm9t",
  "venue": "arXiv",
  "versions": []
}