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Transactions of the American Mathematical Society

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Planar convex bodies, Fourier transform, lattice points, and irregularities of distribution

Authors: L. Brandolini, A. Iosevich and G. Travaglini
Journal: Trans. Amer. Math. Soc. 355 (2003), 3513-3535
MSC (2000): Primary 42B10; Secondary 52A10
Published electronically: April 25, 2003
MathSciNet review: 1990161
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Abstract: Let $B$ be a convex body in the plane. The purpose of this paper is a systematic study of the geometric properties of the boundary of $B$, and the consequences of these properties for the distribution of lattice points in rotated and translated copies of $\rho B$ ($\rho$ being a large positive number), irregularities of distribution, and the spherical average decay of the Fourier transform of the characteristic function of $B$. The analysis makes use of two notions of ``dimension'' of a convex set. The first notion is defined in terms of the number of sides required to approximate a convex set by a polygon up to a certain degree of accuracy. The second is the fractal dimension of the image of the Gauss map of $B$. The results stated in terms of these quantities are essentially sharp and lead to a nearly complete description of the problems in question.

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Additional Information

L. Brandolini
Affiliation: Dipartimento di Ingegneria, Università di Bergamo, Viale G. Marconi 5, 24044 Dalmine (BG), Italy

A. Iosevich
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri

G. Travaglini
Affiliation: Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy

Keywords: Decay of Fourier transforms, convex bodies, Minkowski dimension, lattice points, irregularities of distribution
Received by editor(s): February 11, 2002
Published electronically: April 25, 2003
Additional Notes: The first and third authors are supported by MURST. The second author is supported by NSF grant DMS00-87339
Article copyright: © Copyright 2003 American Mathematical Society

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