{
  "id": "2010.14724",
  "version": "v1",
  "published": "2020-10-28T03:12:40.000Z",
  "updated": "2020-10-28T03:12:40.000Z",
  "title": "Spectrality of generalized Sierpinski-type self-affine measures",
  "authors": [
    "Jing-Cheng Liu",
    "Ying Zhang",
    "Zhi-Yong Wang",
    "Ming-Liang Chen"
  ],
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "For an expanding integer matrix $M\\in M_2(\\mathbb{Z})$ and an integer digit set $D=\\{(0,0)^t,(\\alpha_1,\\alpha_2)^t,(\\beta_1,\\beta_2)^t\\}$ with $\\alpha_1\\beta_2-\\alpha_2\\beta_1\\neq0$, let $\\mu_{M,D}$ be the Sierpinski-type self-affine measure defined by $\\mu_{M,D}(\\cdot)=\\frac{1}{3}\\sum_{d\\in D}\\mu_{M,D}(M(\\cdot)-d)$. In [5.36], the authors separately investigated the spectral property of the measure $\\mu_{M,D}$ in the case of $\\det(M)\\notin 3\\mathbb{Z}$ or $\\alpha_1\\beta_2-\\alpha_2\\beta_1\\notin 3\\mathbb{Z}$. In this paper, we consider the remaining case where $\\det(M)\\in 3\\mathbb{Z}$ and $\\alpha_1\\beta_2-\\alpha_2\\beta_1\\in 3\\mathbb{Z}$, and give the necessary and sufficient conditions for $\\mu_{M,D}$ to be a spectral measure. This completely settles the spectrality of the Sierpinski-type self-affine measure $\\mu_{M,D}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-28T03:12:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "generalized sierpinski-type self-affine measures",
      "spectrality",
      "integer digit set",
      "expanding integer matrix",
      "spectral property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}