Bohr Almost Periodic Sets of Toral Type

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