On the topological structure of univoque sets (Q863965)

The authors study topological properties of the set of real numbers in \((1,2)\) that are univoque (recall that a real number \(\beta\in(1,2)\) is called univoque if 1 has a unique \(\beta\)-expansion \(1=\sum_{i\geq 1} c_i\beta^{-i}\), with \(c_i\in\{0,1\})\). Paralleling the combinatorial characterization of the set \(U\) of univoque numbers [\textit{P. Erdős, I. Joó} and \textit{V. Komornik}, Bull. Soc. Math. Fr. 118, No. 3, 377--390 (1990; Zbl 0721.11005)], the authors give a combinatorial characterization of \(\overline u\), the topological closure of \(u\). In particular \((\overline u\setminus u)\) is a countable dense subset of \(\overline u\). Hence \(\overline u\) is perfect (thus a Cantor set as it is of measure zero). The proof uses a fine study of the combinatorics of words appearing in the expansions in non-integer (univoque) bases.