Bohr Almost Periodic Sets of Toral Type

A locally finite multiset (A, c), A c R, c : A F {1, b} defines a Radon measure p, := Ex, c(k) S. that is Bohr almost periodic in the sense of Favorov if the convolution v. * f is Bohr almost periodic for every f E C). If it is of toral type: the Fourier transform equals zero outside of a rank m < rx) subgroup, then there exists a compactification Mfr : 'r of, a foliation of 'IC`, and a pair (K,K) where K := OA) and K is a measure supported on K such that K = II) 0 where -0' : 711 H R is the Pontryagin dual of *. For (A, c) uniformly discrete, we prove that every connected component of K is homeomorphic to rirm embedded transverse to the foliation and the homotopy of its embedding is a rank m rf subgroup S of 272, and we compute its density as a function of S and *. For n = 1 and K, a nonsingular real algebraic variety, this construction gives all Fourier quasicrystals recently characterized by Olevskii and Ulanovskii.

Automated summary

This passage introduces a locally finite multiset and the Radon measure defined by it. It discusses the Bohr almost periodicity of the measure in the sense of Favorov. It presents a proof of existence for the toral type case and computes the density. Additionally, the passage mentions a construction that can be used to study Fourier quasicrystals.