Fourier Quasicrystals on ${\mathbb{R}}^n$

Abstract

This paper has three aims. First, for $n\ge 1$ 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.

Numerous misprints corrected. Previous version page 25, line 36 Equaation (83) replaced by (69) and Equation (86)replaced by (70) References updated