On the discrepancy of (n\(\alpha\) ). II (Q1073086)

The author gives strong results on the asymptotic behaviour of the discrepancy of the sequence (n\(\alpha)\) for a certain class of \(\alpha\) 's. Using a fundamental result of his recent paper [Acta Arith. 44, 241- 279 (1984; Zbl 0506.10031)] he first gives (possibly) best lower estimates for arbitrary irrational \(\alpha\) (with an absolute constant not depending on \(\alpha)\) and then also gives sharp upper estimates for \(\alpha =[a_ 0;a_ 1,a_ 2,...],\) such that \(a_ i\) is even for \(i\geq 1.\) If in addition the sequence \((a_ k)\) is Cesarò-bounded then a very explicit formula for the asymptotic behaviour of the discrepancy is obtained. This is a very significant generalization of the earlier interesting special cases \(\alpha =(1+\sqrt{5})/2\) (Dupain) and \(\alpha =\sqrt{2}\) (Dupain, Sós) [see the paper of \textit{L. Ramshaw}, J. Number Theory 13, 138-175 (1981; Zbl 0458.10035)].