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PreprintCC0arXiv

The exponent 1/4 in three complexity bounds: integer bases, continued fractions, and the (3/2)n steering word

  • Ralf Stephan
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Abstract

Let ξ be a real algebraic irrational number and p(n,ξ,b) the number of distinct blocks of length n in its base-b expansion. Bugeaud and Evertse proved in 2008 that p(n,ξ,b)/(n(log n)η) is unbounded for every η<1/11. We prove this for every η<1/4. We also prove the analogous bound for the continued fraction expansion of an algebraic number of degree at least three, with every exponent δ<1/4. Bugeaud announced this statement in 2013 with an unspecified δ>0. Finally, the subword complexity of the steering word of (3/2)n exceeds C k(log k)η infinitely often for every η<1/4 and every C. The three proofs share one count: ε−1 classes times ε−3 subspaces from the quantitative Subspace Theorem of Evertse and Ferretti. The subspaces are then disposed of at the cost of a logarithm, with no further power of ε−1. All three results, and the quantitative Subspace Theorem they rest on, are machine-verified in Lean~4.

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Cite

BibTeXRIS
BibTeX
@misc{stephan2026exponent,
  title = {{The exponent $1/4$ in three complexity bounds: integer bases, continued fractions, and the $(3/2)^n$ steering word}},
  author = {Ralf Stephan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30937},
  eprint = {2610.30937},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30937},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - The exponent 1/4 in three complexity bounds: integer bases, continued fractions, and the (3/2)^n steering word
AU  - Ralf Stephan
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.30937
UR  - https://arxiv.org/abs/2610.30937
AB  - Let ξ be a real algebraic irrational number and p(n,ξ,b) the number of distinct blocks of length n in its base-b expansion. Bugeaud and Evertse proved in 2008 that p(n,ξ,b)/(n(log n)^η) is unbounded for every η<1/11. We prove this for every η<1/4. We also prove the analogous bound for the continued fraction expansion of an algebraic number of degree at least three, with every exponent δ<1/4. Bugeaud announced this statement in 2013 with an unspecified δ>0. Finally, the subword complexity of the steering word of (3/2)^n exceeds C k(log k)^η infinitely often for every η<1/4 and every C. The three proofs share one count: ε^−1 classes times ε^−3 subspaces from the quantitative Subspace Theorem of Evertse and Ferretti. The subspaces are then disposed of at the cost of a logarithm, with no further power of ε^−1. All three results, and the quantitative Subspace Theorem they rest on, are machine-verified in Lean~4.
ER  -