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

Abstract

Bourgain posed the problem of calculating $$\begin{aligned} \Sigma = \sup _{n \ge 1} ~\sup _{k_1 < \cdots < k_n} \frac{1}{\sqrt{n}} \left\| \sum _{j=1}^n e^{2 \pi i k_j \theta } \right\| _{L^1([0,1])}. \end{aligned}$$ It is clear that \(\Sigma \le 1\); 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 \(\Sigma \ge \sqrt{\pi }/2 \approx 0.886\), 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.