The size of exponential sums on intervals of the real line

Abstract

We prove that there is a constant $c>0$ depending only on $M\ge 1$ and $\mu \ge 0$ such that $${\int }_y^{y+a}|g(t)|dt\ge \exp (-c/(a\delta )),a\in (0,\pi ],$$ for every $g$ of the form $$g(t)=\sum _{j=0}^n{a}_j{e}^{i{\lambda }_{j}t},a_j\in \mathbb{C},|a_j|\le M{j}^{\mu },|a_0|=1,n\in \mathbb{N},$$ where the exponents ${\lambda }_j\in \mathbb{C}$ satisfy $$\text{Re}({\lambda }_0)=0,\text{Re}({\lambda }_j)\ge j\delta >0,j=1,2,\dots ,$$ and for every subinterval $[y,y+a]$ of the real line. Establishing inequalities of this variety is motivated by problems in physics.

Fixed minor problems and added a new remark