On series $\sum c_{k}f(kx)$ and Khinchin's conjecture

We prove the optimality of a criterion of Koksma (1953) in Khinchin’s conjecture, settling a long standing open problem in analysis. Using this result, we also give a near optimal condition for the a.e.\ convergence of series ${\sum }_{k=1}^{\mathrm{\infty }}{c}_{k}f(kx)$ for $f\in L^2$.