{
  "abstract": "We study geometric inverse problems for finite sets $A\\subset \\mathbb{Z}^d$ whose Fourier $L^1$ norm, or Wiener norm, \\begin{align*} \\|A\\|:=\\int_{\\mathbb{T}^d}\\Big|\\sum_{a\\in A}e(a\\cdot t)\\Big|dt \\end{align*} is subpolynomial in $|A|$. In particular, we show a variety of geometric phenomena are incompatible with small Fourier $L^1$ norm. Our principal application concerns spherical Freiman models. Suppose $\\|A\\|=|A|^{o(1)}$ and $D\\subseteq A$ is Freiman isomorphic to a set $S$ of lattice points on a sphere; then, $|D|/|A|$ must be polynomially small in $|A|$. Moreover, if $|D|/|A|$ is not too small, almost all of $S$ lies in a small number of spherical caps, and almost all caps have rich additive structure. The proof uses a new mechanism for studying additive structure across subsets of $A$, along with decoupling results of Bourgain-Demeter.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26619",
  "author_total": 2,
  "authors": [
    {
      "name": "Alex Burgin"
    },
    {
      "name": "Junzhe Mao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/23z8sxx39mdqe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26619",
  "fulltext_url": null,
  "id": "23z8sxx39mdqe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26619",
  "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",
      "math.CA"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7214107533",
  "pdf_url": "https://arxiv.org/pdf/2609.26619",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/23z8sxx39mdqe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2609.26619",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26619"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Remarks on the Geometry of Sets in $\\mathbb{Z}^d$ with Small Fourier $L^1$ Norm",
  "title_text": "Remarks on the Geometry of Sets in ℤ^d with Small Fourier L^1 Norm",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/23z8sxx39mdqe",
  "venue": "arXiv",
  "versions": []
}