The zero set Λ of exponential polynomials and the Riesz property of its exponential system E_Λ in L²(0 , T)

Consider the class of exponential polynomials of the form f(z)=∑n=0N(∑k=0mncn,kzk)ehnz,0=h0<h1<h2<⋯<hN,such that the coefficients c_n,k are complex numbers with cn,mn≠0 for all n∈ { 0 , 1 , … , N}. Let Λf={λn,μn}n∈Z be the zero set of f, where {λn,μn}n∈Z:={…,λ-1,…,λ-1⏟μ-1-times,λ0,…,λ0⏟μ0-times,λ1,…,λ1⏟μ1-times,…}.We show that the upper Beurling uniform density of Λ _f is equal to h_N/ (2 π). Then based on results by Avdonin and Moran, we prove that if Λ _f is of neutral type, then there is a family of functions of generalized divided differences, denoted by E(Λ) , belonging to the span of the exponential system EΛ={tkeλnt:n∈Z,k=0,1,…,μn-1}, such that E(Λ) is a Riesz sequence in L²(0 , T) for any T> h_N. If Λ _f is not of neutral type, we show that there exist real numbers u_n satisfying | Re (λ_n+ u_n) | = O(1) , so that the exponential system {tke(λn+un)t:n∈Z,k=0,1,…,μn-1}is a Riesz sequence in L²(0 , T) for any T> h_N.