Fourier quasicrystals and Lagarias’ conjecture

Abstract

J. C. Lagarias (2000) conjectured that if μ \mu is a complex measure on p p -dimensional Euclidean space with a uniformly discrete support and its Fourier transform in the sense of distributions is also a measure with a uniformly discrete support, then the support of μ \mu is a subset of a finite union of translates of some full-rank lattice. The conjecture was proved by N. Lev and A. Olevski (2013) in the case p = 1 p=1 . In the case of an arbitrary p p they proved the conjecture only for a positive measure μ \mu .

Here we show that Lagarias’ conjecture does not always hold and investigate two special types of measure distributions connected with the conjecture.