An Erdős-Wintner theorem for second-order linear recurrent bases

[2608.27745] An Erdős-Wintner theorem for second-order linear recurrent bases Skip to main content Search Submit Donate Log in Mathematics > Number Theory arXiv:2608.27745 (math) [Submitted on 27 Aug 2026] Title: An Erdős-Wintner theorem for second-order linear recurrent bases Authors: Johann Verwee View PDF HTML (experimental) Abstract: Let $a,b$ be integers with $1\le b\le a$, and let \[ G_0=1,\qquad G_1=a+1,\qquad G_{n+2}=aG_{n+1}+bG_n. \] For real-valued functions which are additive in the greedy $G$-digits, we prove a necessary-and-sufficient criterion for the existence of a limiting distribution. The criterion consists of a first-order drift series and a quadratic digit-energy series, and it recovers the Zeckendorf theorem when $a=b=1$. The main difficulty is necessity: a one-step transfer matrix detects all non-maximal digit values, while the maximal digit becomes visible only in a two-step product. A weighted Euclidean norm symmetrizes the untwisted companion matrix and makes both contractions occur at the true Perron scale. Sufficiency follows from a two-dimensional Perron product lemma with square-summable transverse perturbations and a convergent, not necessarily absolutely convergent, Perron drift. The limiting characteristic function admits a scalar infinite-product representation in a neighbourhood of the origin and a global matrix-product representation at arbitrary frequencies. Comments: 19 pages Subjects: Number Theory (math.NT) ; Probability (math.PR) MSC classes: 11A63 (Primary), 60F05 (Secondary) Cite as: arXiv:2608.27745 [math.NT] (or arXiv:2608.27745v1 [math.NT] for this version) https://doi.org/10.48550/arXiv.2608.27745 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Johann Verwee [ view email ] [v1] Thu, 27 Aug 2026 22:12:22 UTC (20 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2026-08 Change to browse by: math math.PR References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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