Irrationality Exponents and Partial Quotient Growth in Continued Fractions
Abstract
Let [a1(x),a2(x),…,an(x),…] be the continued fraction expansion of irrational x∈[0,1), and let qn(x) be the denominator of the n-th convergent. In this paper, we study how the growth rate of an+1(x) on a prescribed logarithmic size interacts with its upper growth rate relative to qn(x). For ψ:ℕ→ℝ≥0 satisfying ψ(n)→ ∞ and α, β∈ [0, ∞], define the joint level set Fα,β:={x∈ [0,1): liminfn→∞(log (an+1(x)))/ψ(n)=α, limsupn→∞(log (an+1(x)))/(log qn(x))=β}. We determine the Hausdorff dimension of Fα,β for all values of $ α$ and β. Our results is related to several earlier results on the metric theory of continued fractions, including those of Bugeaud [Math. Ann. {327} (2003)] on irrationality exponents, Wang--Wu [Adv. Math. {218} (2008)] on the growth of partial quotients, and Song--Tan--Zhang [Nonlinearity {37} (2024)] on the joint distribution of convergence and irrationality exponents.
Related
- Irrationality exponents of logarithms of positive rational numbers
- An elementary treatment of Ramanujan's elementary continued fractions
- Irrationality Exponents of Real CM Elliptic Periods and Powers of the Lemniscate Period
- Irrationality exponents of logarithmic numbers: Values, quotients, and graded approximation
- Eta-Quotient Representations for a Three Parameter Family of Modular Functions Associated with the Rogers-Ramanujan Continued Fraction
BibTeX
@misc{cheng2026irrationality,
title = {{Irrationality Exponents and Partial Quotient Growth in Continued Fractions}},
author = {Wanjin Cheng and Jing Feng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25812},
eprint = {2609.25812},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25812},
}
RIS
TY - UNPB
TI - Irrationality Exponents and Partial Quotient Growth in Continued Fractions
AU - Wanjin Cheng
AU - Jing Feng
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25812
UR - https://arxiv.org/abs/2609.25812
AB - Let [a_1(x),a_2(x),…,a_n(x),…] be the continued fraction expansion of irrational x∈[0,1), and let q_n(x) be the denominator of the n-th convergent. In this paper, we study how the growth rate of a_n+1(x) on a prescribed logarithmic size interacts with its upper growth rate relative to q_n(x). For ψ:ℕ→ℝ_≥0 satisfying ψ(n)→ ∞ and α, β∈ [0, ∞], define the joint level set F_α,β:={x∈ [0,1): liminf_n→∞(log (a_n+1(x)))/ψ(n)=α, limsup_n→∞(log (a_n+1(x)))/(log q_n(x))=β}. We determine the Hausdorff dimension of F_α,β for all values of $ α$ and β. Our results is related to several earlier results on the metric theory of continued fractions, including those of Bugeaud [Math. Ann. {327} (2003)] on irrationality exponents, Wang--Wu [Adv. Math. {218} (2008)] on the growth of partial quotients, and Song--Tan--Zhang [Nonlinearity {37} (2024)] on the joint distribution of convergence and irrationality exponents.
ER -