The L1-norm of exponential sums in Zd
Abstract
Let A be a finite set of integers and FA its exponential sum. McGehee, Pigno & Smith and Konyagin have independently proved that the L1-norm of FA 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 L1-norm of exponential sums of sets in the d-dimensional grid Zd. We show that the L1-norm of FA is considerably larger than log|A| when A is a subset of Zd 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.
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