On the closure of characters and the zeros of entire functions

Abstract

The problem to be studied in this paper concerns the closure properties on an interval of a set of characters {e~nx}~, where A = {2n}~ is a given set of real or complex numbers without finite point of accumulation. This problem is for obvious reasons depending on the distribution of zeros of certain entire functions of exponential type. The main problem of the paper is to determine the closure radius Q = Q(A)defined as the upper bound of numbers r such that (ei~x)~EA span the space L 2 ( r , r ) . The value of r does not change if a finite number of points are removed from or adjoined to A. Nor does Q(A) change if the metric in the previous definition is replaced by any other LV-metric, or by a variety of other topologies. I f A contains complex numbers we shall always assume (1)< 6~t ~ (0.1) 9 ~eA ~