Sets of convergence of exponential series

Abstract

THEOREM. Let { a, } denote a sequence of numbers contained in the interval [0, 27r] and dense in a subinterval of length q for some positive number r. Let J(q) denote the set of integers which are divisible by [27r/fl] where the symbol [x ] denotes the largest positive integer not greater than x. If M(77) is a prescribed subset of J(r) then there exists a series (1) that diverges for each integer k in M(77) and converges for each integer k in J(7) -M(77).