The missing proof of Paley's theorem about lacunary coefficients

TL;DR

This paper provides a novel direct proof of Paley's theorem on lacunary coefficients without relying on analytic factorization, extending its application to the Littlewood conjecture on exponential sums in $L^1$ norms.

Contribution

It introduces a new proof technique for Paley's theorem that bypasses analytic factorization, enabling broader applications.

Findings

First direct proof of Paley's theorem without analytic factorization

Extension of Paley's theorem to $L^1$ norms of exponential sums

Resolution of the Littlewood conjecture in this context

Abstract

We modify the standard proof of Paley's theorem about lacunary coefficients of functions in $H^1$ to work without analytic factorization. This leads to the first direct proof of the extension of Paley's theorem that we applied to the former Littlewood conjecture about $L^1$ norms of exponential sums.