Convergence of \(\sum c_kf(k x)\) and the \(\mathrm{Lip}\alpha\) class (Q2846744)

Using more symbols in the title than one usually does, the author succeeds to make the title self-explanatory. What is studied is the convergence almost everywhere (as an extension of the trigonometric case, where Carleson solved Luzin's problem) of the series \(\sum\limits_{k=1}^\infty c_k f(kx)\), for \(1\)-periodic \(f\) with mean zero and from Lip\(\alpha\). The main result of this paper asserts that this is the case for \(\alpha\in[1/4,1/2)\) provided \(\sum\limits_{k=1}^\infty c_k^2 \exp(\frac{2\log k}{\log\log k})\). For these \(\alpha\) the latter condition seriously improves previous results by Gaposhkin, whose results are still among the best.