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

For a periodic function $f$ with bounded variation and integral zero on its period interval, we show that $\sum_{k=1}^\infty c_k^2 (\log\log k)^\gamma <\infty$,$\gamma>4$impliesthealmosteverywhereconvergenceof$\sum_{k=1}^\infty c_k f(n_kx)$foranyincreasingsequence$(n_k)$ 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.