Upper and lower fluctuations in Shallit's law of leap years in Pierce expansions
Abstract
Shallit's law of leap years in Pierce expansions gives, for Lebesgue-almost every x∈[0,1], the upper and lower limits of a normalized difference between Nx and the number of leap years up to the year N determined by the Pierce expansion digits of x. In this paper, we show that, for every x∈[0,1], both limits are determined by the lower limit of a normalized logarithm of the product of the first n digits of x. In particular, the upper and lower limits always have the same absolute value. As applications, we characterize the sets of points with prescribed upper and lower limits, and show that each of these sets, if non-empty, is dense in [0,1] and its intersection with any non-empty open subset of [0,1] has full Hausdorff dimension. We also compare these sets with the sets defined by the growth rate of the digits, and show that, for each finite parameter, their difference has full Hausdorff dimension.
Related
- Uniform lower bounds for the average length of finite Pierce expansions
- Baire-category transfer between Engel and Pierce expansions
- Explicit upper and lower bounds on certain functions related to ζ(s) for s>1
- Upper and lower bounds for heat kernels with very strong degeneracy
- Lower and Upper Lie Nilpotency Indices of Modular Group Algebras
BibTeX
@misc{ahn2026upper,
title = {{Upper and lower fluctuations in Shallit's law of leap years in Pierce expansions}},
author = {Min Woong Ahn},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30919},
eprint = {2609.30919},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30919},
}
RIS
TY - UNPB
TI - Upper and lower fluctuations in Shallit's law of leap years in Pierce expansions
AU - Min Woong Ahn
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.30919
UR - https://arxiv.org/abs/2609.30919
AB - Shallit's law of leap years in Pierce expansions gives, for Lebesgue-almost every x∈[0,1], the upper and lower limits of a normalized difference between Nx and the number of leap years up to the year N determined by the Pierce expansion digits of x. In this paper, we show that, for every x∈[0,1], both limits are determined by the lower limit of a normalized logarithm of the product of the first n digits of x. In particular, the upper and lower limits always have the same absolute value. As applications, we characterize the sets of points with prescribed upper and lower limits, and show that each of these sets, if non-empty, is dense in [0,1] and its intersection with any non-empty open subset of [0,1] has full Hausdorff dimension. We also compare these sets with the sets defined by the growth rate of the digits, and show that, for each finite parameter, their difference has full Hausdorff dimension.
ER -