On convergence and growth of sums $\sum c_{k}f(kx)$

Abstract

For a periodic function $f$ with bounded variation and integral zero on its period interval, we show that ${\sum }_{k=1}^{\infty }{c}_k^2(\log \log k{)}^{\gamma }<\infty$, $\gamma >4$ implies the almost everywhere convergence of ${\sum }_{k=1}^{\infty }{c}_{k}f(n_{k}x)$ for any increasing sequence $(n_k)$ of integers. We also construct an example showing that the previous condition is not sufficient for $\gamma <2$. Finally we give an a.e. bound for the growth of sums ${\sum }_{k=1}^{N}f(n_{k}x)$ differing from the corresponding optimal result for trigonometric sums by a loglog factor.