On the cardinality of \(\beta\)-expansions of some numbers (Q2816450)

Let \(\beta>1\) and denote by \(\sum_\beta(x)\) resp. \(\sum_{\beta,n}(x)\) the set of all \(\beta\)-expansions of \(x\) resp. the set of \(n\)-prefixes of all \(\beta\)-expansions of \(x\) (\(0\leq x\leq \lfloor \beta \rfloor/(\beta -1)\)). The authors show, under a certain finiteness condition on the left shifts on \(\sum_\beta(x)\), that \(\# \sum_\beta(x)=2^{\aleph_0}\) is equivalent to \(\dim_H \sum_\beta(x) >0\) and to \(\lim_{n\rightarrow \infty}\frac{1}{n} \log \# \sum_{\beta,n}(x)>0\). The proof is based on a graph-directed construction which relates an infinite path in a graph to a specific \(\beta\)-expansion. The paper ends with four examples where this construction is made explicit.