×

On the distribution of the sequence \((n\alpha)\) with transcendental \(\alpha\). (English) Zbl 0886.11046

Let \(\gamma^*(\alpha)= \limsup_{N\to\infty} ND^*_N(\alpha)/\log N\), where \(D^*_N(\alpha)\) denotes the star-discrepancy of the sequence \((\{n\alpha\})^\infty_{n=1}\) for an irrational \(\alpha\) of bounded density; i.e. \(\sum^m_{i= 1}a_i= O(m)\), where \(\alpha= [a_0,a_1,a_2,\dots]\) in continued fraction expansion. Y. Dupain and V. T. Sos [in: Topics in classical number theory, Colloq. Budapest 1981, Vol. I, Colloq. Math. Soc. János Bolyai 34, 355-387 (1984; Zbl 0546.10046)] have shown that \(\inf\gamma^*(\alpha)= \gamma^*(1+\sqrt 2)\), where \(B\) is the set of numbers of bounded density. J. Schoißengeier [Math. Ann. 296, 529-545 (1993; Zbl 0786.11043)] expressed \(\gamma^*(\alpha)\) in terms of the continued fraction expansion of \(\alpha\). In the present paper, it is shown that \[ \gamma^*(B^t)= \gamma^*(B^u)= [\gamma^*(1+\sqrt 2),\infty), \] where \(B^t= \{\alpha\in B\): \(\alpha\) is transcendental} und \(B^u= \{\alpha\in B\): \(\alpha\) is a \(U_2\)-number}. Furthermore, this result is extended to numbers \(\alpha\) with partial denominators \(a_i\geq b\) (for all \(i\geq 1\)), where \(b\geq 4\) is some given even integer.
Reviewer: R.F.Tichy (Graz)

MSC:

11K31 Special sequences
11K38 Irregularities of distribution, discrepancy
11J81 Transcendence (general theory)
PDFBibTeX XMLCite
Full Text: DOI EuDML