Besicovitch covering numbers for $\mathcal{B}$-free and other shifts

Abstract

For a finite alphabet $A$ define by $d_1(x,y):={\mathrm{limsup}}_{n\to \infty }\frac{1}{2n+1}\#\{|i|\le n:x_i\ne y_i\}$ the Besicovitch pseudo-metric on $A^{\mathbb{Z}}$. It is well known that a closed subshift of $A^{\mathbb{Z}}$ has finite covering numbers w.r.t. $d_1$ if and only if it is mean-equicontinuous. Here we study, more generally, the scaling behavior of these covering numbers for individual orbits which are generic for an ergodic measure $\mu$ on $A^{\mathbb{Z}}$ with discrete spectrum, and we explore their usefulness as invariants for block code equivalence. We illustrate this by developing tools to determine these covering numbers for various classes of $\mathcal{B}$-free numbers (in particular also for square-free numbers), and we provide a continuous family of measures ${\mu }_s$, all with the same discrete spectrum generated by a single number, but such that ${\mu }_s$- and ${\mu }_{s^{\prime }}$-typical $x$ resp. $x^{\prime }\in A^{\mathbb{Z}}$ have sufficiently different growth of covering numbers such that there are no finite block codes mapping $x\to x^{\prime }$ and $x^{\prime }\to x$. (Indeed, both orbits have different amorphic complexities.)