Baire-category transfer between Engel and Pierce expansions
Abstract
In this paper, we construct an explicit correspondence between Engel and Pierce expansion digit sequences, obtained by adding n−2 to the nth Engel expansion digit, and show that it induces a homeomorphism from the set of irrationals in the unit interval onto the complement of a countable dense subset of this set. This yields a transfer principle for Baire category between the two expansions, and we determine exactly which Borel classes are preserved. We then identify this homeomorphism with the composition of two maps considered by Moroz (2027), and study the continuity of the second map, which sends modified Engel expansions to Pierce expansions. We determine its set of discontinuities, show that every discontinuity is a jump, and prove that the map is nowhere monotone. In particular, this confirms, for this map, the nowhere monotonicity and the continuity outside the exceptional set conjectured by Moroz (2027). As applications, we characterize the non-increasing position-dependent weights for which the associated series of Pierce expansion digits diverges on a comeager set, and we relate the convergence exponents of the Engel and Pierce expansion digit sequences.
Related
- Upper and lower fluctuations in Shallit's law of leap years in Pierce expansions
- Uniform lower bounds for the average length of finite Pierce expansions
- Beta-expansions induced by general Parry polynomials and sharp iteration asymptotics of their transfer operators
- Pierce-Birkhoff conjecture is false
- Lie methods for countably categorical Engel groups: the Wilson conjecture for 4-Engel 5-groups
BibTeX
@misc{ahn2026bairecategory,
title = {{Baire-category transfer between Engel and Pierce expansions}},
author = {Min Woong Ahn},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00946},
eprint = {2610.00946},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00946},
}
RIS
TY - UNPB
TI - Baire-category transfer between Engel and Pierce expansions
AU - Min Woong Ahn
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00946
UR - https://arxiv.org/abs/2610.00946
AB - In this paper, we construct an explicit correspondence between Engel and Pierce expansion digit sequences, obtained by adding n−2 to the nth Engel expansion digit, and show that it induces a homeomorphism from the set of irrationals in the unit interval onto the complement of a countable dense subset of this set. This yields a transfer principle for Baire category between the two expansions, and we determine exactly which Borel classes are preserved. We then identify this homeomorphism with the composition of two maps considered by Moroz (2027), and study the continuity of the second map, which sends modified Engel expansions to Pierce expansions. We determine its set of discontinuities, show that every discontinuity is a jump, and prove that the map is nowhere monotone. In particular, this confirms, for this map, the nowhere monotonicity and the continuity outside the exceptional set conjectured by Moroz (2027). As applications, we characterize the non-increasing position-dependent weights for which the associated series of Pierce expansion digits diverges on a comeager set, and we relate the convergence exponents of the Engel and Pierce expansion digit sequences.
ER -