Approximation orders of real numbers in beta-dynamical systems

Abstract

For any real numbers x∈[0,1] and β>1, denote by Sn(x,β) the partial sum of the first n terms in the β-expansion of x. It is known that for any β>1 and almost all x∈[0,1], or for any x∈(0,1] and almost all β>1, the approximation order of x by Sn(x,β) is β−n. Let φ:N→R+ be a positive function. In this paper, we study the Hausdorff dimensions of the following two setsAβ(φ)={x∈[0,1]:limsupn→∞logβ⁡(x−Sn(x,β))φ(n)=−1},Ax(φ)={β>1:limsupn→∞logβ⁡(x−Sn(x,β))φ(n)=−1}, and complement the dimension theoretic results of these sets in [3], [6] and [18].