The extreme values of local dimension in fractal geometry (Q1002454)

Let \(\mu\) be the probability measure induced by \(S=\sum_{i=0}^\infty q^{-i}X_i\), where \(X_0,X_1,\ldots\) is a sequence of i.i.d. random variables taking values \(0,1,\dots,m\) with equal probability and \(2\leq q\leq m\) are integers. Let \(\alpha(s)\) (resp. \(\alpha_*(s)\)) denote the local (resp. lower local) dimension of \(\mu\) at \(s\in\mathrm{supp}\mu\). The main result states that the infima of \(\alpha(s)\) and \(\alpha_*(s)\) over \(s\in\mathrm{supp}\mu\) coincide and are equal to \[ \alpha_*=\frac{\log(m+1)-\log(r+\sqrt{r^2+4(l+1)})+\log 2} {\log q} \] for \(rq\leq m<rq+r\); \(r=1,\dots,q-1\) and \(l=m-rq\).