Irrationality exponent and convergence exponent in continued fraction expansions

Abstract

Abstract Let x ∈ ( 0 , 1 ) be an irrational number with continued fraction expansion [ a 1 ( x ) , a 2 ( x ) , ⋯ , a n ( x ) , ⋯ ] . We give the multifractal spectrum of the irrationality exponent and the convergence exponent of x defined by v ( x ) : = sup { v > 0 : | x − p q | < 1 q v     for infinitely many   ( q , p ) ∈ ℕ × ℤ } and τ x := inf s ⩾ 0 : ∑ n ⩾ 1 a n − s x < ∞ respectively. To be precise, we completely determine the Hausdorff dimension of E α , v = x ∈ 0 , 1 :   τ x = α ,   v x = v for any α ⩾ 0 ,   v ⩾ 2 .