The lattice point discrepancy of a body of revolution: Improving the lower bound by Soundararajan's method

Abstract

For a convex body B in three-dimensional Euclidean space, which is invariant under rotations around one coordinate axis and has a smooth boundary of bounded nonzero curvature, the lattice point discrepancy (number of integer points minus volume) of a linearly enlarged copy of B is estimated from below. On the basis of a recent method of K. Soundararajan, an Omega-bound is obtained that improves upon all earlier results of this kind.