An upper estimate for the discrepancy of irrational rotations (Q682134)

Let \(\alpha\in (0,1)\) be an irrational number with the continued fraction expansion \(\alpha=[a_1, a_2, \ldots]\), and let \(r_n=p_n/q_n\) be the convergent of \(\alpha\). Ostrowski expansion is \(N=\sum_{j=0}^m b_j q_j\), where \(q_m\leq N<q_{m+1}\), \(0\leq b_0<a_1\), \(0\leq b_j\leq a_{j+1}\), \(j\geq 1\), and \(b_{j-1}=0\) if \(b_j=a_{j+1}\). \textit{J. Schoissengeier} [Acta Arith. 44, 241--279 (1984; Zbl 0506.10031)] gave an upper bound for the discrepancy \(D_N^* (\{i\alpha\})\) \[ ND_{N}^* (\{i\alpha\})\leq C\left(\sum_{n\leq m}a_n+b_m\right), \] where the estimate is optimal except for the constant \(C\). The authors show that \[ ND^*_N(\{i \alpha\})\leq \max \left(\sum_{n:\; odd,\; \leq m}a_n, \sum_{n:\; even, \; \leq m}a_n\right)+b_m+m+1. \]