{
  "id": "1809.02430",
  "version": "v1",
  "published": "2018-09-07T12:14:17.000Z",
  "updated": "2018-09-07T12:14:17.000Z",
  "title": "Arithmetic Progressions with Restricted Digits",
  "authors": [
    "Aled Walker",
    "Alexander Walker"
  ],
  "comment": "11 pages, submitted to American Mathematical Monthly",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "For an integer $b \\geqslant 2$ and a set $S\\subset \\{0,\\cdots,b-1\\}$, we define the Kempner set $\\mathcal{K}(S,b)$ to be the set of all non-negative integers whose base-$b$ digital expansions contain only digits from $S$. These well-studied sparse sets provide a rich setting for additive number theory, and in this paper we study various questions relating to the appearance of arithmetic progressions in these sets. In particular, for all $b$ we determine exactly the maximal length of an arithmetic progression that omits a base-$b$ digit.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-07T12:14:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arithmetic progression",
      "restricted digits",
      "digital expansions contain",
      "kempner set",
      "well-studied sparse sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}