A sequential view of self--similar measures, or, What the ghosts of Mahler and Cantor can teach us about dimension

Abstract

We show that missing $q$-ary digit sets $F\subseteq[0,1]$ have corresponding naturally associated countable binary $q$-automatic sequence $f$. Using this correspondence, we show that the Hausdorff dimension of $F$ is equal to the base-$q$ logarithm of the Mahler eigenvalue of $f$. In addition, we demonstrate that the standard mass distribution $\nu_F$ supported on $F$ is equal to the ghost measure $\mu_f$ of $f$.