{
  "id": "2107.04471",
  "version": "v1",
  "published": "2021-07-09T14:58:30.000Z",
  "updated": "2021-07-09T14:58:30.000Z",
  "title": "Integrability of orthogonal projections, and applications to Furstenberg sets",
  "authors": [
    "Damian Dąbrowski",
    "Tuomas Orponen",
    "Michele Villa"
  ],
  "comment": "25 pages, 3 figures",
  "categories": [
    "math.CA",
    "math.CO",
    "math.MG"
  ],
  "abstract": "Let $\\mathcal{G}(d,n)$ be the Grassmannian manifold of $n$-dimensional subspaces of $\\mathbb{R}^{d}$, and let $\\pi_{V} \\colon \\mathbb{R}^{d} \\to V$ be the orthogonal projection. We prove that if $\\mu$ is a compactly supported Radon measure on $\\mathbb{R}^{d}$ satisfying the $s$-dimensional Frostman condition $\\mu(B(x,r)) \\leq Cr^{s}$ for all $x \\in \\mathbb{R}^{d}$ and $r > 0$, then $$\\int_{\\mathcal{G}(d,n)} \\|\\pi_{V}\\mu\\|_{L^{p}(V)}^{p} \\, d\\gamma_{d,n}(V) < \\infty, \\qquad 1 \\leq p < \\frac{2d - n - s}{d - s}.$$ The upper bound for $p$ is sharp, at least, for $d - 1 \\leq s \\leq d$, and every $0 < n < d$. Our motivation for this question comes from finding improved lower bounds on the Hausdorff dimension of $(s,t)$-Furstenberg sets. For $0 \\leq s \\leq 1$ and $0 \\leq t \\leq 2$, a set $K \\subset \\mathbb{R}^{2}$ is called an $(s,t)$-Furstenberg set if there exists a $t$-dimensional family $\\mathcal{L}$ of affine lines in $\\mathbb{R}^{2}$ such that $\\dim_{\\mathrm{H}} (K \\cap \\ell) \\geq s$ for all $\\ell \\in \\mathcal{L}$. As a consequence of our projection theorem in $\\mathbb{R}^{2}$, we show that every $(s,t)$-Furstenberg set $K \\subset \\mathbb{R}^{2}$ with $1 < t \\leq 2$ satisfies $$\\dim_{\\mathrm{H}} K \\geq 2s + (1 - s)(t - 1).$$ This improves on previous bounds for pairs $(s,t)$ with $s > \\tfrac{1}{2}$ and $t \\geq 1 + \\epsilon$ for a small absolute constant $\\epsilon > 0$. We also prove an analogue of this estimate for $(d - 1,s,t)$-Furstenberg sets in $\\mathbb{R}^{d}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-09T14:58:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "28A78",
      "44A12"
    ],
    "keywords": [
      "furstenberg set",
      "orthogonal projection",
      "applications",
      "integrability",
      "dimensional frostman condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}