The Fourier transform of planar convex bodies and discrepancy over intervals of rotations

This work studies the Fourier transform of the characteristic function of planar convex bodies averaged over affine transformations. We establish lower and upper bounds on the latter quantities in terms of the geometric properties of the bodies considered. The second matter of study is the affine quadratic discrepancy of planar convex bodies, and we present sharp results on its asymptotic behaviour. In particular, we address averages over intervals of rotations, answering an open question of Bilyk and Mastrianni.