Approximation Orders Of A Real Number In A Family Of Beta-Dynamical Systems

In this paper, we study the approximation orders of a real number x ∈ (0, 1) by the partial sums of its β-expansions as β varies in the parameter space {β ∈ ℠: β > 1}. More precisely, letting Sn(x,β) be the partial sum of the first n items of the β-expansion of x, we prove that for any real number x ∈ (0, 1), the approximation order of x by Sn(x,β) is β−n for Lebesgue almost all β > 1. Moreover, we obtain the size of the set of β > 1 for which x can be approximated with a more general order β−φ(n), where φ: ℕ → ℠+ is a positive function. We also determine the Hausdorff dimension of the set Cφ(α) = β > 1 :lim supn→∞ln(x,β) φ(n) = α, 0 ≤ α ≤∞, where ln(x,β) is the number of the longest consecutive zeros just after the nth digit in the β-expansion of x.