2021/07/22 by Wayne M. Lawton, Lawton, Wayne M.
Mathematics · #43A60 #52C23 #55P15 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:43A60 #msc:52C23 #msc:55P15
paper · pdf · doi:10.48550/arxiv.2107.10611
Talk based partially on a preliminary version of this paper was given on 7 June 2021 in the conference on Complex Approximations, Orthogonal Polynomials and Applications, held at the Sirius Institute, Sochi, Black Sea Coast, Russia. Video is recorded at http://caopa.tilda.ws/program
arxiv created 2021/07/22 · arxiv updated 2021/07/23
A locally finite multiset (Λ,c), Λ⊂ \mathbb Rn, c : Λ→ \1,...,b\ defines a Radon measure μ:= ∑λ∈ Λ c(λ) δλ that is Bohr almost periodic in the sense of Favorov if the convolution μ*f is Bohr almost periodic every f ∈ Cc(\mathbb Rn). If it is of toral type: the Fourier transform \mathfrak F μ equals zero outside of a rank m < ∞ subgroup, then there exists a compactification ψ: \mathbb Rn → \mathbb Tm of \mathbb Rn, a foliation of \mathbb Tm, and a pair (K,κ) where K := ψ(Λ) and κ is a measure supported on K such that \mathfrak F κ= (\mathfrak F μ) ∘ \widehat ψ where \widehat ψ: \widehat \mathbb Tm → \widehat \mathbb Rn is the Pontryagin dual of ψ. If (Λ,c) is uniformly discrete Bohr almost periodic and c = 1, we prove that every connected component of K is homeomorphic to \mathbb Tm-n embedded transverse to the foliation and the homotopy of its embedding is a rank m-n subgroup S of \mathbb Zm, and we compute the density of Λ as a function of ψ 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.