Study of almost everywhere convergence of series by mean of martingale methods

Martingale methods are used to study the almost everywhere convergence of general function series. Applications are given to ergodic series, which improves recent results of Fan (2015), and to dilated series, including Davenport series, which completes results of Gapošhkin (1967) (see also Gapošhkin (1968)). Applications are also given to the almost everywhere convergence with respect to Riesz products of lacunary series.