arXiv:2611.05607v1 Announce Type: new Abstract: Mahler showed that for every irrational $\alpha$ and every base ${b\geq 2}$, there is a single positive integer $n$ such that every digit in $\{0, \dots, b-1\}$ occurs infinitely often in the base-$b$ expansion of $n\alpha$. Furstenberg later observed that dynamical methods can be used to give a different proof. Building on this perspective, we give generalizations of Mahler's theorem in two directions: the multipliers are a syndetic subset of certain multiplicative semigroups, and analogous versions hold in higher dimensions. Our main tool is IC-rigidity, a type of rigidity which is a hybrid of topological and measurable constraints.
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