Fourier quasicrystals and Lagarias' conjecture

TL;DR

This paper disproves Lagarias' conjecture in the general case for complex measures with discrete support and spectrum, while identifying two special cases where the conjecture still holds.

Contribution

It demonstrates the falsity of Lagarias' conjecture in the general setting and establishes two new specific cases where the conjecture remains valid.

Findings

Lagarias' conjecture is false in the general case.

Two special cases confirm the conjecture's validity.

Counterexamples and conditions for validity are provided.

Abstract

J.C.Lagarias (2000) conjectured that if $\mu$ is a complex measure on p-dimensional Euclidean space with a uniformly discrete support and its spectrum (Fourier transform) is also a measure with a uniformly discrete support, then the support of $\mu$ is a subset of a finite union of shifts of some full-rank lattice. The conjecture was proved by N.Lev and A.Olevski (2013) in the case p=1. In the case of an arbitrary p they proved the conjecture only for positive measures. Here we show that Lagarias' conjecture is false in the general case and find two new special cases when assertion of the conjecture is valid.