On the lengths of basic intervals in beta expansions (Q2888611)

For the usual \(\beta\)-expansion \(x=\sum_{n=1}^{\infty}\varepsilon_n(x)\beta^{-n}\) of \(x\in(0,1]\) associated to the \(\beta\)-transformation \(T_{\beta}(x)=\beta x-\lceil\beta x\rceil +1\) this paper studies the associated intervals \(I_n(x)\) comprising the set of points that share the same first \(n\) digits in the \(\beta\)-expansion. The length is related to properties of the \(\beta\)-expansion of \(1\), allowing an exact calculation to be made. This is used to show that the decay in the length of these intervals is multifractal with multifractal spectrum dependent on \(\beta\).