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

Abstract.

For a convex body $\mathcal{B}$ in $\mathbb{R}^3$ which is invariant under rotations around one coordinate axis and has a smooth boundary of bounded nonzero curvature, the lattice point discrepancy $P_\mathcal{B} (t)$ (number of integer points minus volume) of a linearly dilated copy $\sqrt t \mathcal{B}$ is estimated from below. On the basis of a recent method of K. Soundararajan [16], an Ω-bound is obtained that improves upon all earlier results of this kind.