On the gaps between zeros of trigonometric polynomials.

Abstract

We show that for every finite set \(0\notin S\subset \mathbb{Z}^{d}\) with the property \(-S=S\), every real trigonometric polynomial \(f\) on the \(d\) dimensional torus \(\mathbb{T}^{d}=\mathbb{R}^{d}/\mathbb{Z}^{d}\) with spectrum in \(S\) has a zero in every closed ball of diameter \(D\left(S\right)\), where \(D\left(S\right)=\sum _{\lambda \in S}\frac{1}{4||\lambda ||_{2}}\), and investigate tightness in some special cases.