Uniform lower bounds for the average length of finite Pierce expansions

Abstract

For integers 1≤ a≤ b, the length of the finite Pierce expansion of a/b is the number of steps of the iteration r↦ bbmod r which starts at a and ends at zero. Erdős and Shallit (1991) proved that the average of these lengths over 1≤ a≤ b is bounded below by a positive constant times loglog b, and suggested that it is at least of order log b. In this paper, we prove that the average length is at least √(2log b)−3(log b)^1/4−11 for every b≥2. We also show that, for each 0<c<√2, the proportion of numerators a for which the length is less than c√(log b) tends to zero exponentially fast in √(log b), with an explicit rate. Our approach refines the interval-counting argument of Erdős and Shallit by restricting the product of the initial digits. As applications, we obtain similar bounds for reduced fractions and for numerators in a fixed congruence class.

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BibTeX RIS

BibTeX

@misc{ahn2026uniform,
  title = {{Uniform lower bounds for the average length of finite Pierce expansions}},
  author = {Min Woong Ahn},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.26594},
  eprint = {2610.26594},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.26594},
}

RIS

TY  - UNPB
TI  - Uniform lower bounds for the average length of finite Pierce expansions
AU  - Min Woong Ahn
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.26594
UR  - https://arxiv.org/abs/2610.26594
AB  - For integers 1≤ a≤ b, the length of the finite Pierce expansion of a/b is the number of steps of the iteration r↦ bbmod r which starts at a and ends at zero. Erdős and Shallit (1991) proved that the average of these lengths over 1≤ a≤ b is bounded below by a positive constant times loglog b, and suggested that it is at least of order log b. In this paper, we prove that the average length is at least √(2log b)−3(log b)^1/4−11 for every b≥2. We also show that, for each 0<c<√2, the proportion of numerators a for which the length is less than c√(log b) tends to zero exponentially fast in √(log b), with an explicit rate. Our approach refines the interval-counting argument of Erdős and Shallit by restricting the product of the initial digits. As applications, we obtain similar bounds for reduced fractions and for numerators in a fixed congruence class.
ER  -

Abstract

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