Dynamics forβ-shifts and Diophantine approximation

Abstract

Abstract. We investigate the β-expansion of an algebraic number in an algebraic base β. Using tools from Diophantine approximation, we prove several results that may suggest a strong difference between the asymptotic behaviour of eventually periodic expansions and that of non-eventually periodic expansions.

On β-expansions of unity for rational and transcendental numbers β

“…A far reaching generalization of (2) with the same upper bound log M (β)/ log β for a so-called Diophantine exponent of the sequence (x k ) ∞ k=1 was obtained by Adamczewski and Bugeaud in [1]. Both in [1] and in the subsequent paper of Bugeaud [5] the main ingredient is the Subspace Theorem.…”

“…In [1] it was shown that if α ∈ [0, 1] and β > 1 are algebraic numbers and the Rényi β-expansion of α is given by…”

“…The importance of the Rényi β-expansion of unity is explained in [10] (see also [13] and [1]). The complexity of the corresponding sequence X = (x j ) ∞ j=1 for some particular algebraic values of β have been studied by Frougny, Masáková and Pelantová in [7] and [8].…”

Quadratic approximation to automatic continued fractions

“…The Diophantine exponent of a, introduced in [4] and denoted by Dio(a), is the supremum of the real numbers ρ for which there exist arbitrarily long prefixes of a that can be expressed in the form U V x for some real number x and finite words U, V such that |U V x |/|U V | ≥ ρ. It is clear from the definitions that 1 ≤ ice(a) ≤ Dio(a) ≤ +∞ and that the initial critical exponent of an ultimately periodic sequence is infinite.…”

A devil's staircase from rotations and irrationality measures for Liouville numbers

“…In terms of the morphology of d β (1), Blanchard [6] classified real numbers greater than one into five classes C 1 to C 5 . On the other hand, Verger-Gaugry [24] focused on the patterns of the consecutive zeros in d β (1), and defined classes Q (1) 0 , Q (2) 0 , Q (3) 0 , Q 1 and Q 2 . These new points of view are quite different from the usual algebraic one.…”