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PreprintarXiv licencearXiv

On digit-appending walks along linear recurrences

  • Arya Mosahab
  • Sahitya Boyyakar
  • Glenn Bruda
  • Scott Duke Kominers
  • Steven J. Miller
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Abstract

We study digit-appending walks in base b≥ 2 along integer linear recurrences with minimal characteristic polynomials having a unique simple dominant root α that lies outside the unit circle. We prove that if α and b are multiplicatively independent, then no infinite walk can have uniformly bounded appended digit lengths. In contrast, we show that when the appended digit length can be unbounded, the same independence hypothesis implies the existence of infinite walks.

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Cite

BibTeXRIS
BibTeX
@misc{mosahab2026digitappending,
  title = {{On digit-appending walks along linear recurrences}},
  author = {Arya Mosahab and Sahitya Boyyakar and Glenn Bruda and Scott Duke Kominers and Steven J. Miller},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18919},
  eprint = {2610.18919},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18919},
}
RIS
TY  - UNPB
TI  - On digit-appending walks along linear recurrences
AU  - Arya Mosahab
AU  - Sahitya Boyyakar
AU  - Glenn Bruda
AU  - Scott Duke Kominers
AU  - Steven J. Miller
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.18919
UR  - https://arxiv.org/abs/2610.18919
AB  - We study digit-appending walks in base b≥ 2 along integer linear recurrences with minimal characteristic polynomials having a unique simple dominant root α that lies outside the unit circle. We prove that if α and b are multiplicatively independent, then no infinite walk can have uniformly bounded appended digit lengths. In contrast, we show that when the appended digit length can be unbounded, the same independence hypothesis implies the existence of infinite walks.
ER  -