{"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/estimates-on-the-dimension-of-self-similar","title":"Estimates on the dimension of self-similar measures with overlaps","arxiv_id":"2103.01700","date":"2021-03-02","proceeding":null,"authors":["De-Jun Feng","Zhou Feng"],"abstract":"In this paper, we provide an algorithm to estimate from below the dimension of self-similar measures with overlaps. As an application, we show that for any $ \\beta\\in(1,2) $, the dimension of the Bernoulli convolution $ \\mu_\\beta $ satisfies \\[ \\dim (\\mu_\\beta) \\geq 0.9804085,\\] which improves a previous uniform lower bound $0.82$ obtained by Hare and Sidorov \\cite{HareSidorov2018}. This new uniform lower bound is very close to the known numerical approximation $ 0.98040931953\\pm 10^{-11}$ for $\\dim \\mu_{\\beta_3}$, where $ \\beta_{3} \\approx 1.839286755214161$ is the largest root of the polynomial $ x^{3}-x^{2}-x-1$. Moreover, the infimum $\\inf_{\\beta\\in (1,2)}\\dim (\\mu_\\beta) $ is attained at a parameter $\\beta_*$ in a small interval \\[ (\\beta_{3} -10^{-8}, \\beta_{3} + 10^{-8}).\\] When $\\beta$ is a Pisot number, we express $\\dim(\\mu_\\beta)$ in terms of the measure-theoretic entropy of the equilibrium measure for certain matrix pressure function, and present an algorithm to estimate $\\dim (\\mu_\\beta)$ from above as well.","url_abs":"https://arxiv.org/abs/2103.01700v2","url_pdf":"https://arxiv.org/pdf/2103.01700v2.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":"estimates-on-the-dimension-of-self-similar","repo_url":"https://github.com/zfengg/DimEstimate","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}