Discrepancy for convex bodies with isolated flat points

TL;DR

This paper investigates how the discrepancy of the integer lattice with respect to dilated convex bodies with flat points behaves asymptotically, revealing effects of flat point normals and providing estimates based on Fourier analysis.

Contribution

It provides new asymptotic estimates for the discrepancy norm considering flat points with various normal directions, including rational and irrational cases.

Findings

Asymptotic expansion when a flat point has a rational normal.

Discrepancy anomalies when flat points have opposite normals.

Smaller discrepancy for bodies with flat points in generic irrational directions.

Abstract

We consider the discrepancy of the integer lattice with respect to the collection of all translated copies of a dilated convex body having a finite number of flat, possibly non-smooth, points in its boundary. We estimate the $L^{p}$ norm of the discrepancy with respect to the translation variable as the dilation parameter goes to infinity. If there is a single flat point with normal in a rational direction we obtain an asymptotic expansion for this norm. Anomalies may appear when two flat points have opposite normals. When all the flat points have normals in generic irrational directions, we obtain a smaller discrepancy. Our proofs depend on careful estimates for the Fourier transform of the characteristic function of the convex body.