Fourier Quasicrystals on $\mathbb R^n$

Abstract

This paper has three aims. First, for $n \geq 1$ we construct a family of real-rooted trigonometric polynomial maps $P : \mathbb C^n \mapsto \mathbb C^n$ whose divisors are Fourier Quasicrystals (FQ). For $n = 1$ these divisors include the first nontrivial FQ with positive integer coefficients constructed by Kurasov and Sarnak [47, and for $n > 1$ they overlap with Meyer's curved model sets [65] and two-dimensional [66] and multidimensional [67] crystalline measures. We prove that the divisors are FQ by directly computing their Fourier transforms using a formula derived in [50].. Second, we extend the relationship between real-rootedness and amoebas, derived for $n = 1$ by Alon, Cohen and Vinzant [1], to the case $n > 1.$ The extension uses results in [10] about homology of complements of amoebas of algebraic sets of codimension $> 1.$ Third, we prove that the divisors of all uniformly generic real-rooted $P$ are FQ. The proof uses the formula relating Grothendieck residues and Newton polytopes derived by Gelfond and Khovanskii [34]. Finally, we note that Olevskii and Ulanovskii [72] have proved that all FQ with positive integer weights are divisors of real-rooted trigonometric polynomials for $n = 1$ but that the situation for $n > 1$ remains unsolved.