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PreprintCC BY-SA 4.0arXiv

Estimates of Lucas series and expansions in non-integer bases

  • Vilmos Komornik
  • Marco Pedicini
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Abstract

Special Fibonacci and Lucas type expansions proved useful to ``correct'' some theorems on the number of expansions in non-integer bases \cite{BaiKomLor2021, KomLorPed2024, KomLorPed2025}. The proof of a key lemma was non-constructive. It was made constructive in the Fibonacci case by giving a precise estimate of the growth of the partial sums of the series of the inverses of the Fibonacci numbers \cite{KomPed2025}. In this paper we extend this result to the general case.

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Cite

BibTeXRIS
BibTeX
@misc{komornik2026estimates,
  title = {{Estimates of Lucas series and expansions in non-integer bases}},
  author = {Vilmos Komornik and Marco Pedicini},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41658},
  eprint = {2610.41658},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41658},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - Estimates of Lucas series and expansions in non-integer bases
AU  - Vilmos Komornik
AU  - Marco Pedicini
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41658
UR  - https://arxiv.org/abs/2610.41658
AB  - Special Fibonacci and Lucas type expansions proved useful to ``correct'' some theorems on the number of expansions in non-integer bases \cite{BaiKomLor2021, KomLorPed2024, KomLorPed2025}. The proof of a key lemma was non-constructive. It was made constructive in the Fibonacci case by giving a precise estimate of the growth of the partial sums of the series of the inverses of the Fibonacci numbers \cite{KomPed2025}. In this paper we extend this result to the general case.
ER  -