Note on Robins' Conjecture in Dimension Four and Higher

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 $d\ge 4$.