Linear independence of exponentials on the real line

be the upper density. If D < ∞ and the series (1) converges uniformly on compact subsets of the complex plane C then an = 0 for all n, [2, 3, 1]. On the other hand, there are sequences of infinite density, in fact with the quotient in the RHS of (2) growing arbitrarily slowly, such that some series (1) with non-zero coefficients converges to zero uniformly on compact subsets of C, see [2, 3, 1]. With a different notion of linear independence, stated in the beginning, one can obtain very complete results. If one of the exponentials does not belong to the closure of the linear span of the rest, then iΛ is a subset of the zero set of the Fourier transform of a measure with bounded support on the real line. And conversely, if iΛ is the zero set of such a Fourier transform