Sequences, Wedges and Associated Sets of Complex Numbers

Let R = (ri' r 2 , ... ) be a finite or infinite (strictly increasing) sequence of positive integers and let (Wi' W2 , ••• ) be a sequence of wedges in the complex plane. Consider the following problem: Characterize those complex numbers c for which (Ll) for k = 1,2, .... It is shown in [1 J and [4J that, under certain assumptions on the wedges and on the density of the sequence R, the set of all complex numbers satisfying (1.1) for k = = 1, 2, ... is finite. The set itself is not identified there. In this paper we assume that W= WI = W2 = ... , where W is the open wedge W( ct ) (the closed wedge W[ ct J) of width 2ct symmetrically located around the nonnegative real axis. We then discuss the set S(R, ct, n) (S[R, ct, nJ) of nonzero complex numbers c which satisfy (Ll) for k = 1, ... , n, where n is either a positive integer or 00. Section 2 is devoted to the case of finite n. Let W = W(ct). Obviously, W(ct/rn) £; £; S(R, ct, n). We prove a necessary condition (Proposition 2.14) and a sufficient condition (Proposition 2.22) for (1.2) W(ct/rn) = S(R, ct, n) . These results are then combined to obtain a characterization of the case (1.2) (Theorem 2.29). In the case that (1.2) does not hold, we give a necessary condition (Proposition 2.39), a sufficient condition (Proposition 2.53) and a characterization (Theorem 2.63) for (1.3) S(R, ct, n) £; W(ct/rn-I).