On a  stronger form  of Salem-Zygmund's inequality for random trigonometric sums with examples

Summary

By applying the majorizing measure method, we obtain a new estimate  of the supremum of random  trigonometric sums. We show that this estimate is strictly stronger than the well-known Salem-Zygmund's estimate, as well as recent general formulations of it obtained by the author. This improvement is obtained by considering the case when the characters are indexed on  sub-exponentially growing sequences of integers. Several remarkable examples are studied.