Metric theory for continued β-fractions

Abstract

For a fixed real number β>1, let ϵ⁎(1,β)=(ϵ1⁎(1,β),ϵ2⁎(1,β),⋯) be the infinite β-expansion of 1. For any n≥1, we define ℓn(β)=sup⁡{j≥0:ϵn⁎(1,β)=⋯=ϵn+j⁎(1,β)=0} be the maximal length of consecutive 0's in ϵ⁎(1,β) starting from ϵn⁎(1,β). Write A0={β>1:{ℓn(β)}is bounded} and let φ:N→R+ be a positive function. For any fixed β∈A0, we show that the Lebesgue measure of the following setE(φ)={x∈[0,1):an(x)≥φ(n)for infinitely manyn∈N} is null or full according to the convergence or divergence of the series ∑n=1∞1φ(n) respectively, where an(x) denotes the n-th partial quotient of the continued β-fraction expansion of x. We further determine the Hausdorff dimension of the set E(φ). Such measure and dimension results generalize a classical theorem of Borel–Bernstein and a well-known result of Wang–Wu (2008) [28] on regular continued fractions.