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

Bourgain posed the problem of calculating $$\mathrm{\Sigma }=\underset{n\ge 1}{\sup }\underset{k_1<\mathrm{.}\mathrm{.}\mathrm{.}<k_n}{\sup }\frac{1}{\sqrt{n}}\mathrm{\parallel }\sum _{j=1}^n{e}^{2\pi i{k}_j\theta }{\mathrm{\parallel }}_{L^1([0,1])}\mathrm{.}$$ It is clear that $\mathrm{\Sigma }\le 1$; beyond that, determining whether $\mathrm{\Sigma }<1$ or $\mathrm{\Sigma }=1$ 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 $\mathrm{\Sigma }\ge \sqrt{\pi }\mathrm{/}2\approx 0.886$, 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.