The size of exponential sums on intervals of the real line

TL;DR

This paper establishes a lower bound on the integral of certain exponential sums over intervals, with bounds depending on the sum's coefficients and exponents, relevant for physics applications.

Contribution

It provides a new exponential lower bound for integrals of exponential sums with specific growth conditions on coefficients and exponents.

Findings

Lower bound on integral of exponential sums over intervals.

Dependence of bounds on coefficients and exponents.

Applicable to problems in physics.

Abstract

We prove that there is a constant $c > 0$ depending only on $M \geq 1$ and $\mu \geq 0$ such that $$\int_y^{y+a}{|g(t)| \, dt} \geq \exp (-c/(a\delta))\,, a \in (0,\pi]\,,$$ for every $g$ of the form $$g(t) = \sum_{j=0}^n{a_j e^{i\lambda_jt}}, a_j \in {\Bbb C}, \enskip |a_j| \leq Mj^\mu\,, \enskip |a_0|=1\,, \enskip n \in {\Bbb N} \,,$$ where the exponents $\lambda_j \in {\Bbb C}$ satisfy $$\text{\rm Re}(\lambda_0) = 0\,, \qquad \text{\rm Re}(\lambda_j) \geq j\delta > 0\,, j=1,2,\ldots\,,$$ and for every subinterval $[y,y+a]$ of the real line. Establishing inequalities of this variety is motivated by problems in physics.