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

TL;DR

This paper employs martingale techniques to analyze the almost everywhere convergence of various function series, including ergodic, dilated, and lacunary series, advancing existing theoretical results.

Contribution

It introduces martingale methods to improve convergence results for ergodic, dilated, and lacunary series, extending prior research in the field.

Findings

Improves convergence results for ergodic series.

Completes previous results on dilated series, including Davenport series.

Establishes almost everywhere convergence for lacunary series with Riesz products.

Abstract

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 \cite{FanETDS}, and to dilated series, including Davenport series, which completes results of Gaposhkin \cite{Gaposhkin67} (see also \cite{Gaposhkin68}). Application is also given to the almost everywhere convergence with respect to Riesz products of lacunary series.