On the gap problem for the sequence \(\| m\beta \|\) (Q1110572)

This paper contains the proofs of special cases of the results announced in the paper reviewed above (see Zbl 0657.10056): If \(\alpha\) \(\not\in {\mathbb{Q}}\), \(\| m\alpha \| =\min \{| m\alpha -k|\), \(k\in {\mathbb{Z}}\}\), \(M_{\phi}(\alpha)=\{m\in {\mathbb{N}}:\) \(\| m\alpha \| <\phi \}\), \(0<\phi <\| \alpha \|\), \(M_{\phi}(\alpha)\) has at most three different gaps. The length of the (possible) third gap is the sum of the first and the second one. Explicit formulas are given using the continued fraction expansion of \(\alpha\).