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The L1-norm of exponential sums in d

Published online by Cambridge University Press:  16 January 2013

Department of Mathematics, University of Rochester, NY 14627, U.S.A. e-mail:


Let A be a finite set of integers and FA(x) = ∑a∈A exp(2πiax) be its exponential sum. McGehee, Pigno and Smith and Konyagin have independently proved that ∥FA1c 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 d. We show that ∥FA1 is considerably larger than log|A| when Ad has multidimensional structure. We furthermore prove similar lower bounds for sets in , which in a technical sense are multidimensional and discuss their connection to an inverse result on the theorem of McGehee, Pigno and Smith and Konyagin.

Research Article
Copyright © Cambridge Philosophical Society 2013

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