The L¹-norm of exponential sums in Z^d

Let A be a finite set of integers and F_A its exponential sum. McGehee, Pigno & Smith and Konyagin have independently proved that the L^1-norm of F_A is at least c log|A| for some absolute constant c. The lower bound has the correct order of magnitude and was first conjectured by Littlewood. In this paper we present lower bounds on the L^1-norm of exponential sums of sets in the d-dimensional grid Z^d. We show that the L^1-norm of F_A is considerably larger than log|A| when A is a subset of Z^d with multidimensional structure. We furthermore prove similar lower bounds for sets in Z, which in a technical sense are multidimensional and discuss their connection to an inverse result on the theorem of McGehee, Pigno & Smith and Konyagin.