A remark on Liao and Rams' result on distribution of the leading partial quotient with growing speed e^n^1/2 in continued fractions

2016/06/08 by Ma, Liangang
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1606.03344

Abstract

For a real x∈(0,1)∖ℚ, let x=[a₁(x),a₂(x),⋯] be its continued fraction expansion. Denote by T_n(x):= max \a_k(x): 1≤ k≤ n\ the leading partial quotient up to n. For any real α∈(0,∞), γ∈(0,∞), let F(γ,α):=\x∈(0,1)∖ℚ: lim_n→∞\fracT_n(x)e^nγ=α\. For a set E⊂ (0,1)∖ℚ, let dim_H E be its Hausdorff dimension. Recently Lingmin Liao and Michal Rams [LR, Theorem 1.3] show that dim_H F(γ,α) is 1 if r∈(0,1/2), it is 1/2 if r∈(1/2,∞) for any α∈(0,∞). In this paper we show that dim_H F(1/2,α)=1/2 for any α∈(0,∞) following Liao and Rams' method, which supplements their result.

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