A remark on Liao and Rams' result on the distribution of the leading partial quotient with growing speed e(n1/2) in continued fractions

作者：Ma, Liangang^*

来源：Comptes Rendus Mathematique, 2017, 355(7): 734-737.

DOI：10.1016/j.crma.2017.05.012

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For a real x is an element of(0, 1) \ Q, let x = [a(1)(x), a(2)(x), ...] be its continued fraction expansion. Denote by
T-n(x) := max{a(k)(x) : 1 <= k <= n}
the maximum partial quotient up to n. For any real alpha is an element of(0, infinity), gamma is an element of(0, infinity), let
F(gamma, alpha) :={x.(0, 1) \ Q : lim(n -> 8) T-n(x)/e(n gamma)= a}.
For a set E is an element of(0, 1) \ Q, let dim(H)E be its Hausdorff dimension. Recently, Lingmin Liao and Michal Rams showed that
dim(H)F(gamma, alpha) = {1 if gamma is an element of(0,1/2) 1/2 if gamma is an element of(1/2, infinity)
for any alpha is an element of(0, infinity). In this paper, we show that dim(H)F(1/2, alpha) = 1/2 for any alpha is an element of(0, infinity) following Liao and Rams' method, which supplements their result.

• 出版日期2017-7
• 单位滨州学院