On the 'pits effect' of Littlewood and Offord

Asymptotic behaviour of the entire functions f(z) = Sigma(infinity)(n=0) e(2 pi in alpha n) z(n)/n!, with real an is studied. It turns out that the Phragmennn-Lindelof indicator of such a function is always non-negative, unless f( z) = e(az). For a special choice of alpha(n) = alpha n(2) with irrational alpha, the indicator is constant and f has completely regular growth in the sense of Levin and Pfluger. Similar functions of arbitrary order are also considered.