{
  "abstract": "Let $Γ\\subset\\mathbb R^2$ be a compact convex graph and define \\[ T(Γ) = \\inf \\left\\{ t: \\dim_{\\mathrm H}(E)>t \\Longrightarrow |E+Γ|>0 \\text{ for every compact }E\\subset\\mathbb R^2 \\right\\}. \\] For a graph over an interval of positive length the smallest possible value is $T(Γ)=1$. We ask whether this optimal conclusion can hold when pointwise Fourier decay of arclength is unavailable. The answer is yes, even for strictly convex curves. We use the Fourier transform convention $\\widehatν(ξ)=\\int e^{-2πi x\\cdotξ}\\,dν(x)$. We construct a strictly convex Lipschitz graph $Γ$ with $T(Γ)=1$ such that, for every nontrivial subarc $Γ_0$ and every $α>0$, \\[ \\limsup_{|ξ|\\to\\infty} |ξ|^α\\left| \\widehat{H^1|_{Γ_0}}(ξ) \\right|= \\infty. \\] We also give a convex example for which arclength on every nontrivial subarc fails even to be a Rajchman measure. The geometric mechanism behind these examples is a positive curved trace: if $Γ$ contains a positive-length subset of a $C^2$ curve whose curvature is bounded away from zero, then $|E+Γ|>0$ whenever $\\dim_{\\mathrm H}(E)>1$. For a nondegenerate graph this gives $T(Γ)=1$. For convex graphs it implies, in particular, that $T(Γ)=1$ whenever the curvature measure has a nonzero absolutely continuous part. The positive-measure proofs are in physical space and use translated-tube intersections and elementary facts about convex functions. The same overlap estimates give Mattila-type lower bounds for the average lengths of the associated curve projections of neighborhoods under the positive curved-trace hypothesis. We also prove a dimension-one endpoint result for sets with a positive-length rectifiable part and formulate the main remaining question: whether every strictly convex Lipschitz graph has the optimal threshold $T(Γ)=1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28770",
  "author_total": 5,
  "authors": [
    {
      "name": "Alex Iosevich"
    },
    {
      "name": "Zhangze Li"
    },
    {
      "name": "Eyvindur Palsson"
    },
    {
      "name": "Krystal Taylor"
    },
    {
      "name": "Alexia Yavicoli"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cm3e3gapakfqd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28770",
  "fulltext_url": null,
  "id": "cm3e3gapakfqd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28770",
  "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",
      "math.CA"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7204942347",
  "pdf_url": "https://arxiv.org/pdf/2608.28770",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cm3e3gapakfqd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.28770",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28770"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Minkowski sums with convex curves without pointwise Fourier decay",
  "title_text": "Minkowski sums with convex curves without pointwise Fourier decay",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/cm3e3gapakfqd",
  "venue": "arXiv",
  "versions": []
}