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 ∈ ℝ : β > 1 }. More precisely, letting S n (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 S n (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 → ℝ + is a positive function. We also determine the Hausdorff dimension of the set C φ (α) = β > 1 : lim sup n → ∞ l n (x , β) φ (n) = α , 0 ≤ α ≤ ∞ , where l n (x , β) is the number of the longest consecutive zeros just after the n th digit in the β -expansion of x. [ABSTRACT FROM AUTHOR]