{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/on-monochromatic-arithmetic-progressions-in","title":"On monochromatic arithmetic progressions in binary words associated with pattern sequences","arxiv_id":"2204.05287","date":"2022-04-11","proceeding":null,"authors":["Bartosz Sobolewski"],"abstract":"Let $e_v(n)$ denote the number of occurrences of a fixed pattern $v$ 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 pattern $v$ 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.","url_abs":"https://arxiv.org/abs/2204.05287v2","url_pdf":"https://arxiv.org/pdf/2204.05287v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"on-monochromatic-arithmetic-progressions-in","repo_url":"https://github.com/bartoszsobolewski/monochromatic-arithmetic-progressions","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}