We study completeness and frame properties of the system $E(\Lambda,\Gamma):=\{t^k e^{2\pi i \lambda t}: \lambda \in \Lambda, k\in \Gamma\}, \quad \Gamma\subset\mathbb{N}_0 = \{0,1,2,\dots\,\}, \, \Lambda \subset \mathbb{R}.$

Let $X(I)$ be a space of functions supported on $I$, e.g. $X=C(I)$ or $X=L^p(I)$, where $I = [-\sigma, \sigma]$. The radius of completeness of the family $E(\mathbb{Z},\Gamma)$ in the space $X$ is denoted by $ r_X(E(\mathbb{Z},\Gamma)) = \sup\{a\geq0: E(\mathbb{Z},\Gamma)\mbox{ is complete in } X(-a,a)\}.$ It is well-known that:

1) $r_{L^2}(E(\mathbb{Z}, \{0\})) = r_{L^2}(\{e^{2\pi i n t}\}_{n \in \mathbb{Z}}) = r_{C}(\{e^{2\pi i n t}\}_{n \in \mathbb{Z}}) = \frac{1}{2};$

2) $\Gamma = \{0,1,2,\dots, N\}$ then $r_C(E(\mathbb{Z}, \Gamma)) = r_{L^2}(E(\mathbb{Z}, \Gamma)) = \frac{\# \Gamma}{2} = \frac{N+1}{2}.$

One may ask the following question: Is it true that for any $\Gamma \subset \mathbb{N}_{0}$ we have $r_C(E(\mathbb{Z}, \Gamma)) = r_{L^2}(E(\mathbb{Z}, \Gamma))?$ It turns out that in general this is false. More precisely, if $\Gamma$ has "gaps" then the answer depends on $\# \Gamma_{odd}$ and $\# \Gamma_{even}$, where $ \Gamma_{odd} = \Gamma \cap (2\mathbb{Z}+1) \quad \text{and} \quad \Gamma_{even} = \Gamma \cap 2\mathbb{Z}. $

We proved the following theorem [A.~Kulikov, A.~Ulanovskii, I. Z., 2022]: Given a finite set $\Gamma \subset \mathbb{N}_{0}$ satisfying $0 \in \Gamma$. Then $r_{L^2}(E(\mathbb{Z}, \Gamma)) = \frac{\# \Gamma}{2} \quad$ and $\quad r_{C}(E(\mathbb{Z}, \Gamma)) = \begin{cases} \# \Gamma_{odd} + \frac{1}{2} , \text{ if } \# \Gamma_{odd} < \# \Gamma_{even}, \\ \# \Gamma_{even} , \text{ if } \# \Gamma_{odd} \ge \# \Gamma_{even}.\\ \end{cases}$

Our argument is based on a description of certain uniqueness sets for lacunary polynomials.

Bibliography: Aleksei Kulikov, Alexander Ulanovskii, Ilya Zlotnikov, Completeness of Certain Exponential Systems and Zeros of Lacunary Polynomials, (2022), arxiv.org/abs/2210.00504.