Bohr Almost Periodic Sets of Toral Type

Abstract

A locally finite multiset $(\Lambda,c),$ $\Lambda \subset \mathbb R^n, c : \Lambda \rightarrow \{1,...,b\}$ defines a Radon measure $\mu := \sum_{\lambda \in \Lambda} c(\lambda)\, \delta_\lambda$ that is Bohr almost periodic in the sense of Favorov if the convolution $\mu*f$ is Bohr almost periodic every $f \in C_c(\mathbb R^n).$ If it is of toral type: the Fourier transform $\mathfrak F \mu$ equals zero outside of a rank $m < \infty$ subgroup, then there exists a compactification $\psi : \mathbb R^n \rightarrow \mathbb T^m$ of $\mathbb R^n,$ a foliation of $\mathbb T^m,$ and a pair $(K,\kappa)$ where $K := \overline {\psi(\Lambda)}$ and $\kappa$ is a measure supported on $K$ such that $\mathfrak F \kappa = (\mathfrak F \mu) \circ \widehat \psi$ where $\widehat \psi : \widehat {\mathbb T^m} \rightarrow \widehat {\mathbb R^n}$ is the Pontryagin dual of $\psi.$ If $(\Lambda,c)$ is uniformly discrete Bohr almost periodic and $c = 1,$ we prove that every connected component of $K$ is homeomorphic to $\mathbb T^{m-n}$ embedded transverse to the foliation and the homotopy of its embedding is a rank $m-n$ subgroup $S$ of $\mathbb Z^m,$ and we compute the density of $\Lambda$ as a function of $\psi$ and the homotopy of comonents of $K.$ For $n = 1$ and $K$ a nonsingular real algebraic variety, this construction gives all Fourier quasicrystals (FQ) recently characterized by Olevskii and Ulanovskii and suggest how to characterize FQ for $n > 1.$