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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.

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