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 $\sum_{k=1}^\infty c_k f(n_kx)$ 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 $\sum_{k=1}^N f(n_kx)$ differing from the corresponding optimal result for trigonometric sums by a loglog factor.