{
  "id": "2204.05287",
  "version": "v1",
  "published": "2022-04-11T17:46:06.000Z",
  "updated": "2022-04-11T17:46:06.000Z",
  "title": "On monochromatic arithmetic progressions in binary words associated with block-counting functions",
  "authors": [
    "Bartosz Sobolewski"
  ],
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let $e_v(n)$ denote the number of occurrences of a fixed block $v$ of digits in the binary expansion of $n \\in \\mathbb{N}$. In this paper we study monochromatic arithmetic progressions in the class of binary words $(e_v(n) \\bmod{2})_{n \\geq 0}$, which includes the famous Thue--Morse word $\\mathbf{t}$ and Rudin--Shapiro word $\\mathbf{r}$. We prove that the length of a monochromatic arithmetic progression of difference $d \\geq 3$ starting at $0$ in $\\mathbf{r}$ is at most $(d+3)/2$, with equality for infinitely many $d$. Moreover, we compute the maximal length of a monochromatic arithmetic progression in $\\mathbf{r}$ of difference $2^k-1$ and $2^k+1$. For a general block $v$ we provide an upper bound on the length of a monochromatic arithmetic progression of any difference $d$. We also prove other miscellaneous results and offer a number of related problems and conjectures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-11T17:46:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "binary words",
      "block-counting functions",
      "study monochromatic arithmetic progressions",
      "difference",
      "thue-morse word"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}