{
  "abstract": "Let $X\\subset\\mathbb{R}^{d_X}$ and $Y\\subset\\mathbb{R}^{d_Y}$ be open, where $d_X,d_Y\\geq2$ and $1\\leq p\\leq\\min\\{d_X,d_Y\\}$. Let $Φ:X\\times Y\\to\\mathbb R^p$ be a smooth mapping, and suppose that $D_xΦ$ and $D_yΦ$ have rank $p$. For compact sets $A\\subset X$ and $B\\subset Y$, we give lower bounds on $\\dim_H A+\\dim_H B$ ensuring that the pinned configuration set \\[Δ_Φ^y(A)=\\{Φ(x,y):x\\in A\\}\\] has positive Lebesgue measure or nonempty interior for some $y\\in B$. The dimensional thresholds are expressed in terms of uniform $L^2$-Sobolev estimates for the generalized Radon transforms associated with $Φ$. Our results extend the Fourier integral operator approach developed by Greenleaf, Iosevich, and Taylor for unpinned $2$-point configuration problems to the pinned setting. We also prove a nonlinear projection theorem that remains valid even when the Peres-Schlag transversality condition fails. Under the smoothing hypotheses used for the pinned configuration theorem, we obtain lower bounds for the Hausdorff dimensions of the corresponding slices of $A$ by the level sets of $Φ$. Applications include distances from points to hyperplanes, spheres and lines; multiparameter distance sets; and a restricted projection family introduced by Harris.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32088",
  "author_total": 4,
  "authors": [
    {
      "name": "Allan Greenleaf"
    },
    {
      "name": "Eyvindur Ari Palsson"
    },
    {
      "name": "Minh-Quy Pham"
    },
    {
      "name": "Georgios Psaromiligkos"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0p4k7nsvsnvzb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32088",
  "fulltext_url": null,
  "id": "0p4k7nsvsnvzb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32088",
  "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"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7223655096",
  "pdf_url": "https://arxiv.org/pdf/2610.32088",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0p4k7nsvsnvzb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:43Z",
      "native_id": "2610.32088",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32088"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Pinned configuration sets and slicing theorems via Fourier integral operators",
  "title_text": "Pinned configuration sets and slicing theorems via Fourier integral operators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:43Z",
  "url": "https://research.draconitic.com/paper/0p4k7nsvsnvzb",
  "venue": "arXiv",
  "versions": []
}