Equality case in van der Corput's inequality and collisions in multiple lattice tilings

TL;DR

This paper characterizes the equality cases in van der Corput's inequality for multiple lattice tilings, revealing more restrictive conditions than in the classical case, and explores implications for multiple lattice tilings.

Contribution

It provides the first explicit characterization of equality cases for all m ≥ 2 in van der Corput's inequality, extending understanding beyond the m=1 case.

Findings

Explicit characterization of equality cases for all m ≥ 2

Equality cases are more restrictive than for m=1

Implications for multiple lattice tilings

Abstract

Van der Corput's provides the sharp bound vol(C) \le m 2^d on the volume of a d-dimensional origin-symmetric convex body C that has 2m-1 points of the integer lattice in its interior. For m=1, a characterization of the equality case vol(C)= m 2^d is equivalent to the well-known problem of characterizing tilings by translations of a convex body. It is rather surprising that so far, for m \ge 2, no characterization of the equality case has been available, though a hint to the respective characterization problem can be found in the 1987 monograph of Gruber and Lekkerkerker. We give an explicit characterization of the equality case for all m \ge 2. Our result reveals that, the equality case for m \ge 2 is more restrictive than for $m=1$. We also present consequences of our characterization in the context of multiple lattice tilings.