On complete and incomplete exponential systems

Given a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$ 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 $\xi \in {\mathbb{R}}^d$, there exists $1\le j\le d+1$ 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 $\mathrm{\Omega }=B_d$, the unit ball, and when $\mathrm{\Omega }=Q_d$, the unit cube. Given a bounded domain $\mathrm{\Omega }$, we say that $e^{2 \pi i x \cdot a}, e^{2 \pi i x \cdot a’}$are$\phi$-approximatelyorthogonalif$$\mathrm{\mid }{\widehat{\chi }}_{\mathrm{\Omega }}(a-a^{\mathrm{\prime }})\mathrm{\mid }\le \varphi (\mathrm{\mid }a-a^{\mathrm{\prime }}\mathrm{\mid }),a\mathrm{\ne }{a}^{\mathrm{\prime }}$$where$\phi: [0, \infty) \to [0, \infty)$isaboundedmeasurablefunctionthattendsto$0$atinfinity\mathrm{.}Weprovethat$L^2(B_d)$doesnotpossessa$\phi$-approximate orthogonal basis of exponentials for a wide range of functions $\varphi$. 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}).