Systems of rank one, explicit Rokhlin towers, and covering numbers

Abstract

AbstractRotations$$f_\alpha $$fαof the one-dimensional torus (equipped with the normalized Lebesgue measure) by an irrational angle$$\alpha $$αare known to be dynamical systems of rank one. This is equivalent to the property that the covering number$$F^*(f_\alpha )$$F∗(fα)of the dynamical system is one. In other words, there exists a basisBsuch that for arbitrarily highh, an arbitrarily large proportion of the unit torus can be covered by the Rokhlin tower$$(f_\alpha ^kB)_{k=0}^{h-1}$$(fαkB)k=0h-1. AlthoughBcan be chosen with diameter smaller than any fixed$$\varepsilon > 0$$ε>0, it is not always possible to take an interval forBbut this can only be done when the partial quotients of$$\alpha $$αare unbounded. In the present paper, we ask what maximum proportion of the torus can be covered whenBis the union of$$n_B \in {\mathbb {N}}$$nB∈Ndisjoint intervals. This question has been answered in the case$$n_B =1$$nB=1by Checkhova, and here we address the general situation. If$$n_B = 2$$nB=2, we give a precise formula for the maximum proportion. Furthermore, we show that for fixed$$\alpha $$α, the maximum proportion converges to 1 when$$n_B \rightarrow \infty $$nB→∞. Explicit lower bounds can be given if$$\alpha $$αhas 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.