On an example by Poincaré and sums with Kronecker sequence

This short and simple communication is motivated by recent papers by L. Colzani and A. Kochergin. We give a brief analysis of an example by Poincaré related to sums of the type ${\sum }_{k=0}^{t-1}f(k\alpha +x)$ where $f$ is a continuous periodic function and $\alpha$ is irrationaland its recent generalisations. Most of the constructions under consideration are well-known. In this note, we just wanted to bring all the results together and give a general and improved multi-dimensional formulation of a recent result by A. Kochergin, prove non-existence of a universal continuous function and discuss some of the related results in terms of Diophantine Approximation. In particular, in our opinion smoothness results involving Diophantine exponents $\omega$, $\widehat{\omega }$ and $\widehat{\lambda }$ had never been documented before.