{
  "id": "2310.08771",
  "version": "v1",
  "published": "2023-10-12T23:34:44.000Z",
  "updated": "2023-10-12T23:34:44.000Z",
  "title": "Complex dimensions for IFS with overlaps",
  "authors": [
    "Nikita Sidorov"
  ],
  "comment": "4 pages, no figures",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "The notion of {\\it complex dimension} of a one-dimensional Cantor set $C=\\bigcap_{n=1}^\\infty C_n$ dates back decades [3]. It is defined as the set of poles of the meromorphic $\\zeta$-function $\\zeta(s)=\\sum_{n=1}^{\\infty}d_j^s$, where $\\Re s>0$, and $d_j$ is the length of the $j$th interval in $C_n$. Following the trend, I switch from sets to measures, which will allow me to generalize the construction to iterated function schemes that do not necessarily satisfy the Open Set Condition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-12T23:34:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "complex dimension",
      "one-dimensional cantor set",
      "open set condition",
      "th interval",
      "iterated function schemes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}