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

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 rhoB (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