Completeness of Certain Exponential Systems and Zeros of Lacunary Polynomials

Let $\mathrm{\Gamma }$ be a subset of $\{0,1,2,\mathrm{.}\mathrm{.}\mathrm{.}\}$. We show that if $\mathrm{\Gamma }$ has `gaps’ then the completeness and frame properties of the system $\{t^k{e}^{2\pi int}:n\in \mathbb{Z},k\in \mathrm{\Gamma }\}$ differ from those of the classical exponential systems. This phenomenon is closely connected with the existence of certain uniqueness sets for lacunary polynomials.