Note on Robins' Conjecture in Dimension Four and Higher
Oleg Asipchuk
TL;DR
The paper addresses whether two convex, centrally symmetric bodies that are not multi-tilers become identical up to rigid motion whenever their Fourier transforms agree on the lattice. It builds a two-stage counterexample: first a planar pair $R$ and $H$ with identical lattice Fourier data, then lifts to higher dimensions by Cartesian product with a ball to obtain $\mathcal{P}=R\times B$ and $\mathcal{Q}=H\times B$, which share $\widehat{1}$ on $\mathbb{Z}^d$ but are not congruent and neither tiles; this disproves the conjecture for $d\ge 4$. The paper also shows that dropping convexity allows counterexamples in low dimensions, highlighting the necessity of convexity for the conjecture to hold. Overall, the results settle the claim negatively in dimensions $d\ge 4$ and delineate the role of convexity in lattice-Fourier rigidity phenomena for tiling.
Abstract
This article is motivated by a conjecture proposed by Sinai Robins in 2024. The conjecture asserts that two convex, centrally symmetric sets of positive measure that are not multi-tilers must coincide up to rigid motions if and only if their Fourier transforms agree on the lattice $\mathbb{Z}^d$. In this paper, we disprove the conjecture by constructing explicit counterexamples in dimensions $d \geq 4$.