On an example by Poincaré and sums with Kronecker sequence

This paper is motivated by recent papers by L. Colzani and A. Kochergin. First of all we discuss an example by Poincaré related to sums of the type \( \sum _{k=0}^{t-1} f(\alpha k +x)\) ${\sum }_{k=0}^{t-1}f(\alpha k+x)$ where f is a continuous function and \(\alpha \) $\alpha$ is an irrational number. Then we deal with its generalisations related to multidimensional Kronecker sequence.