How to write integers in a non-integral basis (Q909694)

Let \(d=(d_ n)\), \(n\geq 0\), be a strictly increasing sequence of natural numbers such that \(d_ 0=1\). Then every natural number \(N\) can be written in a unique manner as \(N=\sum^{n}_{j=0}m_ jd_ j\). Then \(L(d)\) denotes the set of all admissible blocks \(m_ n...m_ 1m_ 0\) obtained as expansions of natural numbers. It is shown that \(L(d)\) is the set of all admissible blocks of digits with respect to a \(\beta\)-expansion if and only if \(m_ n...m_ 1m_ 0\in L(d)\) implies \(m_ n...m_ 1m_ 00\in L(d)\). In this case \(\beta =\lim_{n\to \infty} d_{n+1}d_ n^{-1}.\)