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ISSN 1088-6850(e) ISSN 0002-9947(p)

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

Author(s): L. Brandolini; A. Iosevich; G. Travaglini
Journal: Trans. Amer. Math. Soc. 355 (2003), 3513-3535.
MSC (2000): Primary 42B10; Secondary 52A10
Posted: April 25, 2003
MathSciNet review: 1990161
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Abstract: Let be a convex body in the plane. The purpose of this paper is a systematic study of the geometric properties of the boundary of , 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 . 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 . 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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