PreprintCC BY 4.0arXiv
Asymptotic Behavior of Iterated Sets of Remainders
Abstract
For a positive integer n, let S0(n)={1,2,…,⌊ n/2⌋}, Sj+1(n)={nbmod k:k∈ Sj(n)∖{0}}, and put sj(n) := |Sj(n)|. The sets defined above arise naturally in the study of the length of the Pierce series expansion of a rational number. In \cite{Ba-Vu}, Baraskar and Vukusic conjectured that for every fixed j≥ 2, the limit limn→∞ sj(n)/n does not exist. In this paper, we prove this conjecture in the affirmative. Moreover, we define a new class of iterated remainder sets Tj(n) that naturally arises from the study of the length of the Engel series expansion of a rational number. We analogously study the asymptotic behavior of tj(n) := |Tj(n)|. We show that limn→∞tj(n)/n exists precisely for j∈{0,1} and fails to exist for every fixed integer j≥2.
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BibTeX
@misc{baraskar2026asymptotic,
title = {{Asymptotic Behavior of Iterated Sets of Remainders}},
author = {Omkar Baraskar and Prashant Gokhale and Sarvagya Jain and Adam Kieżun},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24412},
eprint = {2609.24412},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24412},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Asymptotic Behavior of Iterated Sets of Remainders
AU - Omkar Baraskar
AU - Prashant Gokhale
AU - Sarvagya Jain
AU - Adam Kieżun
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24412
UR - https://arxiv.org/abs/2609.24412
AB - For a positive integer n, let S_0(n)={1,2,…,⌊ n/2⌋}, S_j+1(n)={nbmod k:k∈ S_j(n)∖{0}}, and put s_j(n) := |S_j(n)|. The sets defined above arise naturally in the study of the length of the Pierce series expansion of a rational number. In \cite{Ba-Vu}, Baraskar and Vukusic conjectured that for every fixed j≥ 2, the limit lim_n→∞ s_j(n)/n does not exist. In this paper, we prove this conjecture in the affirmative. Moreover, we define a new class of iterated remainder sets T_j(n) that naturally arises from the study of the length of the Engel series expansion of a rational number. We analogously study the asymptotic behavior of t_j(n) := |T_j(n)|. We show that lim_n→∞t_j(n)/n exists precisely for j∈{0,1} and fails to exist for every fixed integer j≥2.
ER -