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Concentration of the Integral Norm of Idempotents

  • Aline Bonami
  • Szilárd Gy. Révész
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Summary

This chapter is a companion of a recent paper, entitled Integral concentration of idempotent trigonometric polynomials with gaps. New results of the present work concern L 1 concentration, while the above-mentioned paper deals with L p concentration. Our aim here is twofold. First we try to explain methods and results, and give further straightforward corollaries. On the other hand, we also push forward the methods to obtain a better constant for the possible concentration (in the L 1 norm) of an idempotent on an arbitrary symmetric measurable set of positive measure. We prove a rather high level γ1 > 0. 96, which contradicts strongly the conjecture of Anderson et al. that there is no positive concentration in the L 1 norm. The same problem is considered on the group \(\mathbb{Z}/q\mathbb{Z}\), with q, say, a prime number. There, the property of absolute integral concentration of idempotent polynomials fails, which is in a way a positive answer to the conjecture mentioned above. Our proof uses recent results of B. Green and S. Konyagin on the Littlewood problem.

Keywords

Positive Measure Trigonometric Polynomial Diophantine Approximation Good Constant Integral Concentration 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

Acknowledgements

The Szilárd Gy. Révész was supported in part by the Hungarian National Foundation for Scientific Research, Project Nos. T-049301, K-61908, and K-72731, and also by the European Research Council, Project ERC-AdG No. 228005.

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Fédération Denis Poisson, MAPMO-UMR 6628 CNRSUniversité d’OrléansOrléansFrance
  2. 2.A. Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary

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