| Popis: |
For a real x ∈ ( 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 α ∈ ( 0 , ∞ ) , γ ∈ ( 0 , ∞ ) , let F ( γ , α ) : = { x ∈ ( 0 , 1 ) ∖ Q : lim n → ∞ T n ( x ) e n γ = α } . For a set E ⊂ ( 0 , 1 ) ∖ Q , let dim H E be its Hausdorff dimension. Recently, Lingmin Liao and Michal Rams showed that dim H F ( γ , α ) = { 1 i f γ ∈ ( 0 , 1 / 2 ) 1 / 2 i f γ ∈ ( 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. |
