Mean Oscillation Modulus and Number-Theoretic Grid Quadrature Formulas

For arbitrary Riemann integrable functions $f$ and irrational numbers $\theta \in (0,1)$, we obtain estimates of the error $R_n(f,\theta)$ of the quadrature formula
$$ \int_{0}^{1}f(x)\,dx=\frac{1}{n}\sum_{k=1}^nf(\{k\theta\})- R_n(f,\theta) $$
in which $\{k\theta\}$ is the fractional part of the number $k\theta$.