  We present new completeness conditions for exponential systems on the complex
plane in Banach algebras of continuous functions on a compact with a connected
complement that are simultaneously holomorphic in the interior of this compact
if it is nonempty, as well as in spaces of holomorphic functions on a simply
connected bounded domain with a topology of uniform convergence on compact
subsets. These conditions are formulated in terms of the Euclidean area of the
convex hull of a compact or a region on the one hand, and new characteristics
of the distribution of exponential system indicators on the other.
