Invariant subspaces with zero density spectrum

In the paper we show that each analytic solution of a homogeneous convolution equation with the characteristic function of minimal exponential type is represented by a series of exponential polynomials in its domain. This series converges absolutely and uniformly on compact subsets in this domain. It is known that if the characteristic function is of minimal exponential type, the density of its zero set is equal to zero. This is why in the work we consider the sequences of exponents having zero density. We provide a simple description of the space of the coefficients for the aforementioned series. Moreover, we provide a complete description of all possible system of functions constructed by rather small groups, for which the representation by the series of exponential polynomials holds.