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