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

Brandolini, L.; Iosevich, A.; Travaglini, G.

doi:10.1090/S0002-9947-03-03240-9

Planar convex bodies, Fourier transform, lattice points, and irregularities of distribution
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by L. Brandolini, A. Iosevich and G. Travaglini PDF

Trans. Amer. Math. Soc. 355 (2003), 3513-3535 Request permission

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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