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The L1-norm of exponential sums in d


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.

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[1] A. Balog and I. Z. Ruzsa A new lower bound for the L1 mean of exponential sums with the Möbius function. Bull. Lond. Math. Soc. 31 (1999), 415418.

[2] P. J. Cohen On a conjecture of Littlewood and idempotent measures. Amer. J. Math. 82 (1960), 191212.

[4] B. J. Green and T. Sanders Boolean functions with small spectral norm. Geom. Funct. Anal. 18 (1) (2008), 144162.

[5] B. J. Green and T. Sanders A quantitative version of the idempotent theorem in harmonic analysis. Ann. of Math. (2) 168 (3) (2008), 10251054.

[8] L. Lorch The principal term in the asymptotic expansion of the Lebesgue constants. Amer. Math. Monthly 61 (1954), 245249.

[9] O. C. McGehee , L. Pigno and B. Smith Hardy's inequality and the L1 norm of exponential sums. Ann. of Math.(2) 113 (3) (1981), 613618.

[12] R. C. Vaughan The L1 mean of exponential sums over primes. Bull. Lond. Math. Soc. 20 (2) (1988), 121123.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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