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The L1-norm of exponential sums in d

Published online by Cambridge University Press:  16 January 2013

GIORGIS PETRIDIS
GIORGIS PETRIDIS*
Affiliation:
Department of Mathematics, University of Rochester, NY 14627, U.S.A. e-mail: giorgis@cantab.net
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Abstract

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 ∥FA∥1 ≥ 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 d. We show that ∥FA∥1 is considerably larger than log|A| when A ⊂ d 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 154 , Issue 3 , May 2013 , pp. 381 - 392
Copyright
Copyright © Cambridge Philosophical Society 2013

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References

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