Univoque numbers and an avatar of Thue–Morse

Abstract

Univoque numbers are real numbers λ > 1 such that the number 1 admits a unique expansion in base λ, i.e., a unique expansion 1 = j≥0 a j λ −(j+1) , with a j ∈ {0, 1, . . . , ⌈λ⌉ − 1} for every j ≥ 0. A variation of this definition was studied in 2002 by Komornik and Loreti, together with sequences called admissible sequences. We show how a 1983 study of the first author gives both a result of Komornik and Loreti on the smallest admissible sequence on the set {0, 1, . . . , b}, and a result of de Vries and Komornik (2007) on the smallest univoque number belonging to the interval (b, b + 1), where b is any positive integer. We also prove that this last number is transcendental. An avatar of the Thue-Morse sequence, namely the fixed point beginning in 3 of the morphism 3 → 31, 2 → 30, 1 → 03, 0 → 02, occurs in a "universal" manner.

Hausdorff dimension of unique beta expansions

“…Then t 1 ≥ ⌈(N − 1)/2⌉. By Definition 2.1 of an admissible block one can directly verify that (t 1 · · · t p ) ∞ ≥ (t 1 t 1 ) ∞ (see also [3,Proposition 2]). Note by (2)…”

Intersections of homogeneous Cantor sets and beta-expansions

“…have the same length β ℓ+1 and all gaps (called (ℓ+1)-level gaps) between them have the same length 1]) and the right endpoint of φ J(N −1) ([0, 1]) coincides with the right endpoint of φ J ([0, 1]). The requirement β ∈ (1/(2N − 1), 1/N ) implies the following simple properties:…”

“…For a more general digit set Ω, there also exist some results on the smallest admissible sequence which is related to the Thue-Morse sequence (cf. [1]).…”

“…We will be interested in the self-similar set Γ β,Ω generated by an IFS {φ d (·) : d ∈ Ω}, where Ω is a finite set of integers, and 1 for some N ≥ 2 and β ∈ (0, 1/N ). It is well known that one can establish a surjective map π Ω : Ω ∞ → Γ β,Ω by letting (1) π…”

Factor and Palindromic Complexity of Thue-Morse’s Avatars

“…The main result of this paper is the computation of the factor and palindromic complexity of two infinite words which appear in [1] as a representation of some significant univoque numbers. A real number λ > 1 is said to be univoque if 1 admits a unique expansion in base λ of the form 1 = i>0 a i λ −i with a i ∈ {0, 1, .…”