Exact Diophantine approximation of real numbers by $ \beta $-expansions

Abstract

We introduce the exact approximation order in the dynamics of βexpansions which has its analogy in classic Diophantine approximation. More precisely, let E β (ψ) be the set of real numbers in [0, 1) which are approximable by their convergents in β-expansions to order ψ but to no better order. The Hausdorff dimension of E β (ψ) is given for any monotonic function ψ.