On the discrepancy of the sequence \((n\alpha)\) (Q1903503)

Let \(\alpha\) be an irrational number given in continued fraction expansion \(\alpha= [a_0, a_1, a_2, \dots ]\) and let \(D^*_N (\alpha)\) denote the star-discrepancy of the sequence \((n \alpha)^\infty_{n=1} \bmod 1\). The author studies properties of the map \[ \alpha\mapsto \nu^* (\alpha):= \limsup_{N\to \infty} {{ND^*_N (\alpha)} \over {\log N}}. \] It is well known that any sequence satisfies \(ND^*_N (x_n)= \Omega (\log N)\) and that the discrepancy of \(D^*_N (\alpha)= O(\log N)\) if and only if \(\alpha\) is of bounded density. Furthermore it is known that \(\nu^* (\alpha)\) usually grows when the partial denominators of \(\alpha\) increase, the minimum is attained at \(\sqrt {2}-1\); see \textit{Y. Dupain} and \textit{V. T. Sos} [Topics in classical number theory, Vol. 1, Colloq. Math. Soc. Janos Bolyai 34, 355-387 (1984; Zbl 0546.10046)]. \textit{J. Schoissengeier} established an explicit formula for \(\nu^* (\alpha)\); cf. [Math. Ann. 296, 529-545 (1993; Zbl 0786.11043)]. In the present paper the author studies the image of the map \(\alpha\mapsto \nu^* (\alpha)\). The main result is \(\nu^* (\mathbb{R} \setminus \mathbb{Q})= [\nu^*( \sqrt {2}- 1), \infty]\). Furthermore it is shown that ``small changes'' in the continued fraction expansion of \(\alpha\) do not change \(\nu^* (\alpha)\).