Let that \(\Omega\subset \Bbb R^d\) be a bounded domain with positive Lebesgue measure. Assume that \(\mathcal F\subset L^2(\Omega)\) is a non-empty set. We say \(\mathcal F\) is an exponentially complete system of functions, or is simply exponentially complete, if for all \(x\in\Bbb R^d\), there is a function \(f\in \mathcal F\) such that \(\hat f(\xi):= \int_\Omega f(x) e^{-2\pi i x\cdot \xi} dx\neq 0.\)

Motivated by the problem of the presence of orthogonal bases of exponential functions, this talk focuses on the cases where \(\mathcal F=\mathcal E(A)\) is a system of exponential functions, i.e, \(\mathcal E(A):=\{f_a(x):=e^{2\pi i x\cdot a}: ~ a\in A\subset \Bbb R^d\},\) and when \(\Omega\subset \Bbb R^d\) is the unit ball \(d>1\) or is \(Q_d\) is the unit cube. To be more specific, we will investigate the maximum (minimum) size of \(A\) for which \(\mathcal E(A)\) is exponentially complete (incomplete) with respect to the ambient space dimension. This is a joint work with Alex Iosevich.