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

TL;DR

This paper investigates the scaling behavior of Besicovitch covering numbers for orbits in symbolic dynamical systems, especially for $eta$-free numbers, and explores their role as invariants for block code equivalence.

Contribution

It introduces a method to analyze Besicovitch covering numbers for $eta$-free numbers and demonstrates their effectiveness as invariants distinguishing orbits under block codes.

Findings

Covering numbers grow differently for various $eta$-free numbers.

Distinct measures with same spectrum can produce non-equivalent orbits.

Orbits with different amorphic complexities cannot be related by finite block codes.

Abstract

For a finite alphabet $A$ define by $d_1(x,y):=\limsup_{n\to\infty}\frac{1}{2n+1}\#\{|i|\le n: x_i\neq y_i\}$ 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 $\mu_{s'}$-typical $x$ resp. $x'\in A^{\mathbb Z}$ have sufficiently different growth of covering numbers such that there are no finite block codes mapping $x\to x'$ and $x'\to x$. (Indeed, both orbits have different amorphic complexities.)