{
  "id": "2212.00340",
  "version": "v1",
  "published": "2022-12-01T07:58:20.000Z",
  "updated": "2022-12-01T07:58:20.000Z",
  "title": "Spectrality of a class of infinite convolutions on $\\mathbb{R}$",
  "authors": [
    "Sha Wu",
    "Yingqing Xiao"
  ],
  "categories": [
    "math.CA",
    "math.DS"
  ],
  "abstract": "Given an integer $m\\geq1$. Let $\\Sigma^{(m)}=\\{1,2, \\cdots, m\\}^{\\mathbb{N}}$ be a symbolic space, and let $\\{(b_{k},D_{k})\\}_{k=1}^{m}:=\\{(b_{k}, \\{0,1,\\cdots, p_{k}-1\\}t_{k}) \\}_{k=1}^{m}$ be a finite sequence pairs, where integers $| b_{k}| $, $p_{k}\\geq2$, $|t_{k}|\\geq 1$ and $ p_{k},t_{1},t_{2}, \\cdots, t_{m}$ are pairwise coprime integers for all $1\\leq k\\leq m$. In this paper, we show that for any infinite word $\\sigma=\\left(\\sigma_{n}\\right)_{n=1}^{\\infty}\\in\\Sigma^{(m)}$, the infinite convolution $$ \\mu_{\\sigma}=\\delta_{b_{\\sigma_{1}}^{-1} D_{\\sigma_{1}}} * \\delta_{\\left(b_{\\sigma_{1}} b_{\\sigma_{2}}\\right)^{-1} D_{\\sigma_{2}}} * \\delta_{\\left(b_{\\sigma_{1}} b_{\\sigma_{2}} b_{\\sigma_{3}}\\right)^{-1}D_{\\sigma_{3}}} * \\cdots $$ is a spectral measure if and only if $p_{\\sigma_n}\\mid b_{\\sigma_n}$ for all $n\\geq2$ and $\\sigma\\notin \\bigcup_{l=1}^\\infty\\prod_{l}$, where $\\prod_{l}=\\{i_{1}i_{2}\\cdots i_{l}j^{\\infty}\\in\\Sigma^{(m)}: i_{l}\\neq j, |b_{j}|=p_{j}, |t_{j}|\\neq1\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-01T07:58:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A25",
      "28A80",
      "42C05",
      "46C05"
    ],
    "keywords": [
      "infinite convolution",
      "spectrality",
      "finite sequence pairs",
      "pairwise coprime integers",
      "infinite word"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}