Irrationality exponent and convergence exponent in continued fraction expansions

Abstract

Let x∈(0,1) be an irrational number with continued fraction expansion [a1(x),a2(x),⋯,an(x),⋯] . We give the multifractal spectrum of the irrationality exponent and the convergence exponent of x defined by v(x):=sup{v>0:|x−pq|<1qv for infinitely many (q,p)∈ℕ×ℤ} and τx:=infs⩾0:∑n⩾1an−sx<∞ respectively. To be precise, we completely determine the Hausdorff dimension of Eα,v=x∈0,1: τx=α, vx=v for any α⩾0, v⩾2 .