On sums of fractional parts (Q6595581)

Let \N\[\NS(\alpha,n) = \sum_{k=1}^n \left( \{k \alpha\} - \frac{1}{2} \right), \N\]\Nwhere \(\{ \cdot \}\) denotes the fractional part. The author proves the following theorem:\N\NTheorem: Assume that \(\alpha\) is irrational and \(\alpha+\beta\) is rational. Then the values of \(|S(\alpha,n) + S(\beta,n)|\) are unbounded in \(n\) unless \(\alpha + \beta \in \mathbb{Z}\), in which case the value is zero.\N\NThis problem is related to problems in lattice point counting, and to the study of so-called bounded remainder sets in the theory of uniform distribution modulo 1 (such as the classical work of \textit{H. Kesten} [Acta Arith. 12, 193--212 (1966; Zbl 0144.28902)]). The proof uses results on the local discrepancy of \((n \alpha)\)-sequences due to \textit{I. Oren} [Isr. J. Math. 42, 353--360 (1982; Zbl 0533.28009)] and \textit{J. Schoissengeier} [Acta Arith. 133, No. 2, 127--157 (2008; Zbl 1228.11105)].