Completeness of exponential systems in functional spaces in terms of perimeter

A new scale of completeness conditions for exponential systems is established for two types of functional spaces on subsets of the complex plane. The first type of spaces are Banach spaces of functions that are continuous on a compact set and holomorphic in the interior of this compact set (if it is nonempty) with the uniform norm. The second type consists of spaces of holomorphic functions on a bounded open set with the topology of uniform convergence on compact sets. These conditions are formulated in terms of majorizing the perimeter of the convex hull of the domain of functions from the space by new characteristics of the distribution of exponents of the exponential system.