{
  "id": "2303.14731",
  "version": "v1",
  "published": "2023-03-26T14:22:15.000Z",
  "updated": "2023-03-26T14:22:15.000Z",
  "title": "Quantization dimension for inhomogeneous bi-Lipschitz IFS",
  "authors": [
    "Amit Priyadarshi",
    "Mrinal K. Roychowdhury",
    "Manuj Verma"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\nu$ be a Borel probability measure on a $d$-dimensional Euclidean space $\\mathbb{R}^d$, $d\\geq 1$, with a compact support, and let $(p_0, p_1, p_2, \\ldots, p_N)$ be a probability vector with $p_j>0$ for $1\\leq j\\leq N$. Let $\\{S_j: 1\\leq j\\leq N\\}$ be a set of contractive mappings on $\\mathbb R^d$. Then, a Borel probability measure $\\mu$ on $\\mathbb R^d$ such that $\\mu=\\sum_{j=1}^N p_j\\mu\\circ S_j^{-1}+p_0\\nu$ is called an inhomogeneous measure, also known as a condensation measure on $\\mathbb R^d$. For a given $r\\in (0, +\\infty)$, the quantization dimension of order $r$, if it exists, denoted by $D_r(\\mu)$, of a Borel probability measure $\\mu$ on $\\mathbb R^d$ represents the speed at which the $n$th quantization error of order $r$ approaches to zero as the number of elements $n$ in an optimal set of $n$-means for $\\mu$ tends to infinity. In this paper, we investigate the quantization dimension for such a condensation measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-26T14:22:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A50",
      "28A80",
      "94A34"
    ],
    "keywords": [
      "quantization dimension",
      "inhomogeneous bi-lipschitz ifs",
      "borel probability measure",
      "condensation measure",
      "dimensional euclidean space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}