On the convergence of $\sum c_{n} f(nx)$ and the Lip 1/2 class

We investigate the almost everywhere convergence of $\sum c_{n} f(nx)$, where $f$ is a measurable function satisfying

By a known criterion, if $f$ satisfies the above conditions and belongs to the Lip $\alpha$ class for some $\alpha > 1/2$, then $\sum c_{n} f(nx)$ is a.e. convergent provided $\sum c_{n}^{2} < +\infty$. Using probabilistic methods, we prove that the above result is best possible; in fact there exist Lip 1/2 functions $f$ and almost exponentially growing sequences $(n_{k})$ such that $\sum c_{k} f(n_{k} x)$ is a.e. divergent for some $(c_{k})$ with $\sum c_{k}^{2} < +\infty$. For functions $f$ with Fourier series having a special structure, we also give necessary and sufficient convergence criteria. Finally we prove analogous results for the law of the iterated logarithm.