Subword complexity and Projection Bodies

Christian Steineder

Technische Universität Wien, Austria

A polytope $P \subset [0,1)^d$ and an $\vec \alpha \in [0,1)^d$ induce a generalized Sturmian sequence ${\bf h }(P,\vec \alpha) \in \{0,1\}^{\mathbb{Z}}$, called Hartman sequence, which is by definition $1$ at the $k$-th position iff $k \vec \alpha$    mod $1 \in P$ and 0 otherwise, $k \in \mathbb{Z}$. We prove an asymptotic formula for the subword complexity of such a Hartman sequence. This result establishes a connection between symbolic dynamics and convex geometry: If the polytope $P$ is convex then the asymptotic complexity of ${\bf h }(P,\vec \alpha)$ equals for almost all $\vec \alpha \in [0,1)^d$ the volume of the projection body $\Pi P$ of $P$.

Supported by FWF project no S9612.