On the distribution of recurrence times in nonlinear system (Q1110571)

Let \(\alpha\) be an irrational number, \(I=[a,b)\subseteq [0,1)\), \(M=M(\alpha,I)=\{m\in {\mathbb{N}},\quad m\alpha \in I\quad (mod 1)\}=\{m_ k:\quad m_ 1<m_ 2<...\}\) then \textit{K. Florek} [Colloq. Math. Wroclaw 2, 323-324 (1951)], stated without proof and \textit{N. B. Slater} [Proc. Camb. Phil. Soc. 63, 1115-1123 (1967; Zbl 0178.047)] proved (for \(a=0)\) that \(2\leq card\{m_{k+1}-m_ k,\quad k\in {\mathbb{N}}\}\leq 3.\) The author announces a generalization of this result to arbitrary subintervals \(I\subseteq {\mathbb{R}}/{\mathbb{Z}}\) (i.e. including the case \(I=[0,a)\cup (b,1)).\) These results can be translated into results about the recurrence times of the Poincaré map of an integrable Hamiltonian system with two degrees of freedom. For (partial) proofs see the paper reviewed below (Zbl 0657.10057).