Universal β-expansions

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Given β∈(1,2), a β-expansion of a real x is a power series in base 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 that for any β∈(1,2) and a.e. x∈ (0,1) there always exists a universal β-expansion of x in the sense of Erdös and Komornik, i.e., a β-expansion whose complexity function is 2ⁿ. We also study some questions related to the points having less than a full branching continuum of β-expansions and also normal β-expansions.