Systems of Rank One, Explicit Rokhlin Towers, and Covering Numbers

TL;DR

This paper investigates how well rotations of the circle can be covered by unions of a fixed number of disjoint intervals, providing formulas and bounds for maximum coverage and exploring the influence of number-theoretic properties.

Contribution

It extends previous work by analyzing the maximum coverage of the torus with multiple intervals and establishes explicit formulas and convergence results.

Findings

Maximum coverage formula for two intervals

Maximum coverage approaches 1 as number of intervals increases

Explicit bounds for constant partial quotients

Abstract

Rotations $f_\alpha$ of the one-dimensional torus (equipped with the normalized Lebesgue measure) by an irrational angle $\alpha$ are known to be dynamical systems of rank one. This is equivalent to the property that the covering number $F^*(f_\alpha)$ of the dynamical system is one. In other words, there exists a basis $B$ such that for arbitrarily high $h$ an arbitrarily large proportion of the unit torus can be covered by the Rokhlin tower $(f_\alpha^kB)_{k=0}^{h-1}$. Although $B$ can be chosen with diameter smaller than any fixed $\varepsilon > 0$, it is not always possible to take an interval for $B$ but this can only be done when the partial quotients of $\alpha$ are unbounded. In the present paper, we ask what maximum proportion of the torus can be covered when $B$ is the union of $n_B \in \mathbb{N}$ disjoint intervals. This question has been answered in the case $n_B =1$ by Checkhova, and here we address the general situation. If $n_B = 2$ we give a precise formula for the maximum proportion. Furthermore, we show that for fixed $\alpha$ the maximum proportion converges to $1$ when $n_B \to \infty$. Explicit lower bounds can be given if $\alpha$ has constant partial quotients. Our approach is inspired by the construction involved in the proof of the Rokhlin Lemma and furthermore makes use of the Three Gap Theorem.