Hausdorff dimensions of certain sets in terms of the Sylvester series

Abstract For x ∈ ( 0 , 1 ] , let x = ∑ n = 1 ∞ 1 d n ( x ) be its Sylvester series with digits { d n ( x ) } n ≥ 1 , { P n ( x ) Q n ( x ) } n ≥ 1 be its S-convergents. We calculate the Hausdorff dimension of the set { x ∈ ( 0 , 1 ] : lim n → ∞ ⁡ log ⁡ d n ( x ) ϕ ( n ) = 1 } , where ϕ is a positive function defined on N . We also consider the set of points with lim n → ∞ ⁡ log ⁡ d n ( x ) ϕ ( n ) = 0 or ∞. In addition, for any v ≥ 0 , we obtain the Hausdorff dimension of the Jarnik-like set { x ∈ ( 0 , 1 ] : | x − P n ( x ) Q n ( x ) | ≤ 1 Q n ( x ) v + 1 for infinitely many n ∈ N } .