Let be a convex body in with piecewise smooth boundary and let denote the Fourier transform of its characteristic function. In this paper we determine the admissible decays of the spherical -averages of and we relate our analysis to a problem in the geometry of convex sets. As an application we obtain sharp results on the average number of integer lattice points in large bodies randomly positioned in the plane.
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Luca Brandolini, Marco Rigoli, Giancarlo Travaglini, Average decay of Fourier transforms and geometry of convex sets. Rev. Mat. Iberoam. 14 (1998), no. 3, pp. 519–560DOI 10.4171/RMI/244