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Exact approximation order of real numbers in Cantor series expansions
Abstract
Let Q={qn}n≥1 be a sequence of integers with qn≥2 for every n≥1. Every x∈[0,1) admits a Cantor series expansion of the form x = ε1(x)/q1 + ε2(x)/q1q2 +⋯+ εn(x)/(q1q2⋯ qn) +⋯.In this paper, we study exact approximation of real numbers by the n-th partial sums of their Cantor series expansions. More precisely, for a non-increasing function ψ:ℕ→(0,∞) satisfying ψ(n)→0, let Ec(ψ) be the set of points that are ψ-approximable by their n-th partial sums but are not bψ-approximable for any 0<b<1. We determine the Hausdorff dimension of Ec(ψ).
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BibTeX
@misc{cheng2026exact,
title = {{Exact approximation order of real numbers in Cantor series expansions}},
author = {Wanjin Cheng and Xinyun Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.30435},
eprint = {2606.30435},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.30435},
}
RIS
TY - UNPB
TI - Exact approximation order of real numbers in Cantor series expansions
AU - Wanjin Cheng
AU - Xinyun Zhang
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2606.30435
UR - https://arxiv.org/abs/2606.30435
AB - Let Q={q_n}_n≥1 be a sequence of integers with q_n≥2 for every n≥1. Every x∈[0,1) admits a Cantor series expansion of the form x = ε_1(x)/q_1 + ε_2(x)/q_1q_2 +⋯+ ε_n(x)/(q_1q_2⋯ q_n) +⋯.In this paper, we study exact approximation of real numbers by the n-th partial sums of their Cantor series expansions. More precisely, for a non-increasing function ψ:ℕ→(0,∞) satisfying ψ(n)→0, let E_c(ψ) be the set of points that are ψ-approximable by their n-th partial sums but are not bψ-approximable for any 0<b<1. We determine the Hausdorff dimension of E_c(ψ).
ER -