{
  "id": "2409.12826",
  "version": "v1",
  "published": "2024-09-19T14:57:59.000Z",
  "updated": "2024-09-19T14:57:59.000Z",
  "title": "Dimension of Diophantine approximation and applications",
  "authors": [
    "Longhui Li",
    "Bochen Liu"
  ],
  "comment": "41 pages",
  "categories": [
    "math.CA",
    "math.CO",
    "math.NT"
  ],
  "abstract": "In this paper we construct a new family of sets via Diophantine approximation, in which the classical examples are endpoints. Our first application is on their Hausdorff dimension. We show a recent result of Ren and Wang, known sharp on orthogonal projections in the plane, is also sharp on $A+cB$, $c\\in C$, thus completely settle this ABC sum-product problem. Higher dimensional examples are also discussed. In addition to Hausdorff dimension, we also consider Fourier dimension. In particular, now for every $0\\leq t\\leq s\\leq 1$ we have an explicit construction in $\\mathbb{R}$ of Hausdorff dimension $s$ and Fourier dimension $t$, together with a measure $\\mu$ that captures both dimensions. In the end we provide a perspective of Knapp's example and treat our Diophantine approximation as its analog in $\\mathbb{R}$, that naturally leads to the sharpness of Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem. These are alternatives of recent examples due to Fraser-Hambrook-Ryou. In particular, to deal with the non-geometric case we construct measures of \"Hausdorff dimension\" $a$ and Fourier dimension $b$, even if $a<b$. This clarifies some difference between sets and measures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-19T14:57:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "diophantine approximation",
      "hausdorff dimension",
      "fourier dimension",
      "mockenhaupt-mitsis-bak-seeger fourier restriction theorem",
      "abc sum-product problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}