{
  "id": "2212.07577",
  "version": "v1",
  "published": "2022-12-15T01:33:48.000Z",
  "updated": "2022-12-15T01:33:48.000Z",
  "title": "Fourier bases of a class of planar self-affine measures",
  "authors": [
    "Ming-Liang Chen",
    "Jing-Cheng Liu",
    "Zhi-Yong Wang"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $\\mu_{M,D}$ be the planar self-affine measure generated by an expansive integer matrix $M\\in M_2(\\mathbb{Z})$ and a non-collinear integer digit set $D=\\left\\{\\begin{pmatrix} 0\\\\0\\end{pmatrix},\\begin{pmatrix} \\alpha_{1}\\\\ \\alpha_{2} \\end{pmatrix}, \\begin{pmatrix} \\beta_{1}\\\\ \\beta_{2} \\end{pmatrix}, \\begin{pmatrix} -\\alpha_{1}-\\beta_{1}\\\\ -\\alpha_{2}-\\beta_{2} \\end{pmatrix}\\right\\}$. In this paper, we show that $\\mu_{M,D}$ is a spectral measure if and only if there exists a matrix $Q\\in M_2(\\mathbb{R})$ such that $(\\tilde{M},\\tilde{D})$ is admissible, where $\\tilde{M}=QMQ^{-1}$ and $\\tilde{D}=QD$. In particular, when $\\alpha_1\\beta_2-\\alpha_2\\beta_1\\notin 2\\Bbb Z$, $\\mu_{M,D}$ is a spectral measure if and only if $M\\in M_2(2\\mathbb{Z})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-15T01:33:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A25",
      "28A80",
      "42C05",
      "46C05"
    ],
    "keywords": [
      "planar self-affine measure",
      "fourier bases",
      "non-collinear integer digit set",
      "spectral measure",
      "expansive integer matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}