Bohr Almost Periodic Sets of Toral Type

Abstract

A locally finite multiset $(\Lambda ,c),$ $\Lambda \subset {\mathbb {R}}^n, c : \Lambda \mapsto \{1,...,b\}$ 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 for 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 $\psi : {\mathbb {R}}^n \mapsto {\mathbb {T}}^m$ 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 ${\widehat{\psi }} : \widehat{{\mathbb {T}}^m} \mapsto \widehat{{\mathbb {R}}^n}$ is the Pontryagin dual of $\psi .$ For $(\Lambda ,c)$ uniformly discrete, 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 its density as a function of S and $\psi .$ For $n = 1$ and K, a nonsingular real algebraic variety, this construction gives all Fourier quasicrystals recently characterized by Olevskii and Ulanovskii.