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.