Calculation of improper integrals using \((n\alpha)\)-sequences (Q1606066)

The authors give a necessary and sufficient condition for the relation \(\frac{1}{N}\sum^{N}_{n=1}f (\{n \alpha\})\rightarrow \int^{1}_{0}f(x) dx\) to hold as \(N \rightarrow\infty.\) Here \(\alpha\) is an irrational number, \(\{x\}\) denotes the fractional part of \(x\) and \(f\) is from a suitable class of functions with the property that \(\lim_{x\rightarrow \beta +}f(x)= \pm \infty\) or \(\lim_{x\rightarrow \beta -}f(x)= \pm \infty\) for a finite set of rational \(\beta \in [0,1].\)