Besicovitch covering numbers for B-free and other shifts

For a finite alphabet A define by d₁(x,y):=n→∞1/2n+1#\|i|≤ n: xᵢ≠ yᵢ\ the Besicovitch pseudo-metric on AZ. It is well known that a closed subshift of AZ has finite covering numbers w.r.t. d₁ 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 $μ$ on AZ 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 B-free numbers (in particular also for square-free numbers), and we provide a continuous family of measures μₛ, all with the same discrete spectrum generated by a single number, but such that μₛ- and μs'-typical x resp. x'∈ AZ have sufficiently different growth of covering numbers such that there are no finite block codes mapping x→ x' and x'→ x. (Indeed, both orbits have different amorphic complexities.)