Universal β-expansions

Given β ∈ (1, 2), a β-expansion of a real x is a series in decreasing powers of β with coefficients 0 and 1 whose sum equals x. The aim of this note is to study certain problems related to the universality and combinatorics of β-expansions. Our main result is as follows: for each β ∈ (1, 2) and a.e. x ∈ (0, 1) there always exists a universal β-expansion of x in the sense of Erdős and Komornik, i.e., a β-expansion whose complexity function is 2. Besides, we study some properties of points having a “small” set of β-expansions and finish the paper by considering normal β-expansions. 1. FORMULATION OF MAIN RESULTS Let β ∈ (1, 2) be our parameter and put Σ = ∞1 {0, 1}. Fix x ≥ 0; we call a sequence ε ∈ Σ a β-expansion of x, if it satisfies (1.1) x = πβ(ε) := ∞ ∑