  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}).
