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

Abstract Rotations $$f_\alpha $$ fα of the one-dimensional torus (equipped with normalized Lebesgue measure) by an irrational angle $$\alpha xmlns:mml="http://www.w3.org/1998/Math/MathML">α are known to be dynamical systems rank one. This is equivalent property that covering number $$F^*(f_\alpha )$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">F∗(fα) system In other words, there exists a basis B such for arbitrarily high h , large proportion unit can covered Rokhlin tower $$(f_\alpha ^kB)_{k=0}^{h-1}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">(fαkB)k=0h-1 . Although chosen diameter smaller than any fixed $$\varepsilon > 0$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">ε>0 it not always possible take interval but this only done when partial quotients unbounded. present paper, we ask what maximum union $$n_B \in {\mathbb {N}}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">nB∈N disjoint intervals. question has been answered in case =1$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">nB=1 Checkhova, and here address general situation. If = 2$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">nB=2 give precise formula proportion. Furthermore, show converges 1 \rightarrow \infty xmlns:mml="http://www.w3.org/1998/Math/MathML">nB→∞ Explicit lower bounds given if constant quotients. Our approach inspired construction involved proof lemma furthermore makes use three gap theorem.