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

Christoph Aistleitner

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 $Σ\leq 1…