On the Result of Invariance of the Closure Set of the Real Projections of the Zeros of an Important Class of Exponential Polynomials

Abstract

We provide the proof of a practical pointwise characterization of the set RP defined by the closure set of the real projections of the zeros of an exponential polynomial P(z)=∑j=1ncjewjz with real frequencies wj linearly independent over the rationals. As a consequence, we give a complete description of the set RP and prove its invariance with respect to the moduli of the cj′s, which allows us to determine exactly the gaps of RP and the extremes of the critical interval of P(z) by solving inequations with positive real numbers. Finally, we analyse the converse of this result of invariance.