On a Conjecture of Littlewood and Idempotent Measures

Abstract

Introduction. In this paper we shall be concerned with two problems. The first is a problem of Littlewood [1] in classical Fourier analysis concerning a lower bound for the LI norm of certain exponential sums. The second is the problem of determining all the idempotent measures on a locally compact abelian group. This second problem we shall solve entirely and thus complete a line of investigation begun by ilelson [3] and Rudin [5]. The problem of idempotent measures is related to the question of describing all homomorphisms of the algebra l' (G) into the algebra L' (H) where G and H are two locally compact abelian groups. We shall treat this problem in a subsequent paper. As will be explained, the problem of Littlewood is closely connected to the problem of idempotent measures, and though the first is stated on the circle group the method of proof will be indispensable for the analysis on the more general class of abelian groups. We now state the problem of Littlewood. Consider an exponential sum