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FREIMAN THEOREM, FOURIER TRANSFORM AND ADDITIVE STRUCTURE OF MEASURES

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  01 February 2009

A. IOSEVICH  and
M. RUDNEV
A. IOSEVICH*
Affiliation:
University of Missouri, Columbia, MO 65211, USA (email: iosevich@math.missouri.edu)
M. RUDNEV
Affiliation:
University of Bristol, Bristol BS8 1TW, UK (email: m.rudnev@bris.ac.uk)
*
✉For correspondence; e-mail: iosevich@math.missouri.edu
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Abstract

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We use the Freiman theorem in arithmetic combinatorics to show that if the Fourier transform of certain measures satisfies sufficiently bad estimates, then the support of the measure possesses an additive structure. The result is then discussed in light of the Falconer distance problem.

Keywords

Fourier transformdecay estimates

MSC classification

Secondary: 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 86 , Issue 1 , February 2009 , pp. 97 - 109
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

This work was partly supported by the grant DMS02-45369 from the National Science Foundation, the National Science Foundation Focused Research Grant DMS04-56306, and the EPSRC grant GR/S13682/01.

References

[1]Brandolini, L., Hofmann, S. and Iosevich, A., ‘Sharp rate of average decay of the Fourier transform of a bounded set’, Geom. Funct. Anal. 13 (2003), 671–680.CrossRefGoogle Scholar
[2]Brandolini, L., Iosevich, A. and Travaglini, G., ‘Spherical L 1-averages of the Fourier transform of the characteristic function of a convex set and the geometry of the Gauss map’, Trans. Amer. Math. Soc. 355 (2003), 3513–3535.CrossRefGoogle Scholar
[3]Brandolini, L., Rigoli, M. and Travaglini, G., ‘Average decay of Fourier transforms and geometry of convex sets’, Rev. Mat. Iberoamericana 14 (1998), 519–560.CrossRefGoogle Scholar
[4]Erdoğan, B., ‘A bilinear Fourier extension theorem and applications to the distance set problem’, Int. Math. Res. Not. IMRN 23 (2005), 1411–1425.CrossRefGoogle Scholar
[5]Erdős, P., ‘On sets of distances of n points’, Amer. Math. Monthly 53 (1946), 248–250.CrossRefGoogle Scholar
[6]Green, B. J., ‘Structure theory of set addition’. ICMS Instructional Conference in Combinatorial Aspects of Mathematical Analysis, Edinburgh, 25 March–5 April 2002.http://www.dpmms.cam.ac.uk/∼bjg23/papers/icmsnotes.pdf.Google Scholar
[7]Green, B. J. and Ruzsa, I., ‘Freiman’s theorem in an arbitrary abelian group’, J. London Math. Soc. 75(1) (2007), 163–175.CrossRefGoogle Scholar
[8]Falconer, K. J., ‘On the Hausdorff dimensions of distance sets’, Mathematika 32 (1985), 206–212.CrossRefGoogle Scholar
[9]Falconer, K. J., The Geometry of Fractal Sets (Cambridge University Press, Cambridge, 1985).CrossRefGoogle Scholar
[10]Freiman, G., Foundations of a Structural Theory of Set Addition, Translations of Mathematical Monographs, 37 (American Mathematical Society, Providence, RI, 1973).Google Scholar
[11]Hart, D., Iosevich, A., Koh, D. and Rudnev, M., ‘Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdős–Falconer distance conjecture’. Preprint arXiv:0707.3473, July 2007.Google Scholar
[12]Iosevich, A., ‘Fourier transform, L 2 restriction theorem, and scaling’, Boll. Unione Mat. Ital. 8 (1999), 383–387.Google Scholar
[13]Iosevich, A. and Łaba, I., ‘K-distance, Falconer conjecture, and discrete analogs’. Integers, Electronic Journal of Combinatorial Number Theory, Proceedings of the Integers Conference in Honor of Tom Brown, pp. 95–106.Google Scholar
[14]Iosevich, A. and Rudnev, M., ‘Non-isotropic distance measures for lattice-generated sets’, Publ. Mat. 49(1) (2005), 225–247.CrossRefGoogle Scholar
[15]Iosevich, A. and Rudnev, M., ‘Erdös distance problem on vector spaces over finite fields’, Trans. Amer. Math. Soc. 359 (2007), 6127–6142.CrossRefGoogle Scholar
[16]Mattila, P., ‘Spherical averages of Fourier transforms of measures with finite energy: dimensions of intersections and distance sets’, Mathematika 34 (1987), 207–228.CrossRefGoogle Scholar
[17]Podkorytov, A., ‘On the asymptotics of the Fourier transform on a convex curve’, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 125 (1991), 50–57.Google Scholar
[18]Shparlinski, I. E., ‘On some generalisations of the Erdős distance problem over finite fields’, Bull. Austral. Math. Soc. 73(2) (2006), 285–292.CrossRefGoogle Scholar
[19]Wolff, T., ‘Decay of circular means of Fourier transforms of measures’, Int. Math. Res. Not. IMRN 10 (1999), 547–567.CrossRefGoogle Scholar