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The size of exponential sums on intervals of the real line

We prove that there is a constant $c > 0$ depending only on $M \geq 1$ and $μ\geq 0$ such that $$\int_y^{y+a}{|g(t)| \, dt} \geq \exp (-c/(aδ))\,, a \in (0,π]\,,$$ for every $g$ of the form $$g(t) = \sum_{j=0}^n{a_j e^{iλ_jt}}, a_j \in {\Bbb C}, \enskip |a_j| \leq Mj^μ\,, \enskip |a_0|=1\,, \enskip n \in {\Bbb N} \,,$$ where the exponents $λ_j \in {\Bbb C}$ satisfy $$\text{\rm Re}(λ_0) = 0\,, \qquad \text{\rm Re}(λ_j) \geq jδ> 0\,, j=1,2,\ldots\,,$$ and for every subinterval $[y,y+a]$ of the real line. Establishing inequalities of this variety is motivated by problems in physics.