On complete and incomplete exponential systems

Given a bounded domain $\Omega \subset {\Bbb R}^d$ with positive measure and a finite set $A=\{a^1, a^2, \dots, a^d\}$, we say that the set ${\mathcal E}(A)={\{e^{2 \pi i x \cdot a^j}\}}_{a^j \in A}$ is a complete exponential system if for every $\xi \in {\Bbb R}^d$, there exists $1 \leq j \leq d+1$ such that \begin{equation} \label{completedef} \int_{\Omega} e^{-2 \pi i x \cdot (a^j-\xi)} dx \not=0; \end{equation} otherwise ${\mathcal E}(A)$ is called an incomplete exponential system. In this paper, we essentially classify complete and incomplete exponential systems when $\Omega=B_d$, the unit ball, and when $\Omega=Q_d$, the unit cube. Given a bounded domain $\Omega$, we say that $e^{2 \pi i x \cdot a}, e^{2 \pi i x \cdot a'}$ are $\phi$-approximately orthogonal if $$|\widehat{\chi}_{\Omega}(a-a')| \leq \phi(|a-a'|), \ a\neq a'$$ where $\phi: [0, \infty) \to [0, \infty)$ is a bounded measurable function that tends to $0$ at infinity. We prove that $L^2(B_d)$ does not possess a $\phi$-approximate orthogonal basis of exponentials for a wide range of functions $\phi$. The proof involves connections with the theory of distances in sets of positive Lebesgue upper density originally developed by Furstenberg, Katznelson and Weiss (\cite{FKW90}).