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

Abstract

Bourgain posed the problem of calculating $$\Sigma =\underset{n\ge 1}{\sup }\underset{k_1<...<k_n}{\sup }\frac{1}{\sqrt{n}}\Vert \sum _{j=1}^n{e}^{2\pi i{k}_j\theta }{\Vert }_{L^1([0,1])}.$$ 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.