On a problem of Bourgain concerning the $L^1$-norm of exponential sums

TL;DR

This paper investigates a problem posed by Bourgain regarding the $L^1$-norm of exponential sums, establishing a new lower bound that advances understanding of spectral types in ergodic theory.

Contribution

The paper proves a new lower bound for Bourgain's supremum problem using a two-dimensional central limit theorem for lacunary series, improving previous results.

Findings

Established that $oxed{rac{ ext{L}^1 ext{-norm}}{ ext{normalization}}} ext{ is at least } rac{ ext{sqrt(pi)}}{2} ext{, i.e., approximately } 0.886.

Used a quantitative two-dimensional CLT for lacunary trigonometric series in the proof.

Improved the known lower bound for Bourgain's problem from previous results.

Abstract

Bourgain posed the problem of calculating $$ \Sigma = \sup_{n \geq 1} ~\sup_{k_1 <... < k_n} \frac{1}{\sqrt{n}}\| \sum_{j=1}^n e^{2 \pi i k_j \theta}\|_{L^1([0,1])}. $$ It is clear that $\Sigma \leq 1$; beyond that, determining whether $\Sigma < 1$ or $\Sigma=1$ would have some interesting implications, for example concerning the problem whether all rank one transformations have singular maximal spectral type. In the present paper we prove $\Sigma \geq \sqrt{\pi}/2 \approx 0.886$, by this means improving a result of Karatsuba. For the proof we use a quantitative two-dimensional version of the central limit theorem for lacunary trigonometric series, which in its original form is due to Salem and Zygmund.