{
  "id": "1611.09762",
  "version": "v1",
  "published": "2016-11-29T18:11:03.000Z",
  "updated": "2016-11-29T18:11:03.000Z",
  "title": "An improved bound on the packing dimension of Furstenberg sets in the plane",
  "authors": [
    "Tuomas Orponen"
  ],
  "comment": "28 pages, 1 figure. This paper overlaps with, and supersedes, arXiv:1610.06745",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $0 \\leq s \\leq 1$. A set $K \\subset \\mathbb{R}^{2}$ is a Furstenberg $s$-set, if for every unit vector $e \\in S^{1}$, some line $L_{e}$ parallel to $e$ satisfies $$\\dim_{\\mathrm{H}} [K \\cap L_{e}] \\geq s.$$ The Furstenberg set problem, introduced by T. Wolff in 1999, asks for the best lower bound for the dimension of Furstenberg $s$-sets. Wolff proved that $\\dim_{\\mathrm{H}} K \\geq \\max\\{s + 1/2,2s\\}$ and conjectured that $\\dim_{\\mathrm{H}} K \\geq (1 + 3s)/2$. The only known improvement to Wolff's bound is due to Bourgain, who proved in 2003 that $\\dim_{\\mathrm{H}} K \\geq 1 + \\epsilon$ for Furstenberg $1/2$-sets $K$, where $\\epsilon > 0$ is an absolute constant. In the present paper, I prove a similar $\\epsilon$-improvement for all $1/2 < s < 1$, but only for packing dimension: $\\dim_{\\mathrm{p}} K \\geq 2s + \\epsilon$ for all Furstenberg $s$-sets $K \\subset \\mathbb{R}^{2}$, where $\\epsilon > 0$ only depends on $s$. The proof rests on a new incidence theorem for finite collections of planar points and tubes of width $\\delta > 0$. As another corollary of this theorem, I obtain a small improvement for Kaufman's estimate from 1968 on the dimension of exceptional sets of orthogonal projections. Namely, I prove that if $K \\subset \\mathbb{R}^{2}$ is a linearly measurable set with positive length, and $1/2 < s < 1$, then $$\\dim_{\\mathrm{H}} \\{e \\in S^{1} : \\dim_{\\mathrm{p}} \\pi_{e}(K) \\leq s\\} \\leq s - \\epsilon$$ for some $\\epsilon > 0$ depending only on $s$. Here $\\pi_{e}$ is the orthogonal projection onto the line spanned by $e$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-29T18:11:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "packing dimension",
      "orthogonal projection",
      "furstenberg set problem",
      "best lower bound",
      "unit vector"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}