Universal \(\beta\)-expansions
For fixed \(\beta\in(1,2)\) and \(x\geq0\) the authors consider \(\beta\)-expansions, i.e., expansions of the form \(x=\sum_{n=1}^\infty\varepsilon_n\beta^{-n}\), where \(\epsilon=(\varepsilon_n)\) is a \(0-1\) sequence. Such an expansion is called universal if every finite \(0-1\) word occurs as an interval in the sequence \(\epsilon\). The author showed earlier [Am. Math. Mon. 110, 838--842 (2003; Zbl 1049.11085)] that almost every \(x\in I_\beta=[0,1/(\beta-1)]\) has a continuum of distinct \(\beta\)-expansions. Now he proves that almost every \(x\in I_\beta\) has a universal \(\beta\)-expansion and poses the question of whether every \(x\) which has a universal \(\beta\)-expansion must have a continuum of distinct \(\beta\)-expansions.
scientific article; zbMATH DE number 2108669
scientific article