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

  • Johann Verwee
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Abstract

Let a,b be integers with 1≤ b≤ a, and let G0=1, G1=a+1, Gn+2=aGn+1+bGn. 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.

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Cite

BibTeXRIS
BibTeX
@misc{verwee2026erdőswintner,
  title = {{An Erdős-Wintner theorem for second-order linear recurrent bases}},
  author = {Johann Verwee},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.27745},
  eprint = {2608.27745},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.27745},
}
RIS
TY  - UNPB
TI  - An Erdős-Wintner theorem for second-order linear recurrent bases
AU  - Johann Verwee
PY  - 2026
DA  - 2026/08/27
PB  - arXiv
DO  - 10.48550/arxiv.2608.27745
UR  - https://arxiv.org/abs/2608.27745
AB  - Let a,b be integers with 1≤ b≤ a, and let G_0=1, G_1=a+1, 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.
ER  -