Pure point measures with sparse support and sparse Fourier–Bohr support

Abstract

Fourier transformable Radon measures are called doubly sparse when both the measure and its transform are pure point measures with sparse support. Their structure is reasonably well understood in Euclidean space, based on the use of tempered distributions. Here, we extend the theory to second countable, locally compact Abelian groups, where we can employ general cut and project schemes and the structure of weighted model combs, along with the theory of almost periodic measures. In particular, for measures with Meyer set support, we characterise sparseness of the Fourier-Bohr spectrum via conditions of crystallographic type, and derive representations of the measures in terms of trigonometric polynomials. More generally, we analyse positive definite, doubly sparse measures in a natural cut and project setting, which results in a Poisson summation type formula.

A note on tempered measures

“…Here, we consider the important special case of measures with uniformly discrete support, for which the three key notions turn out to be equivalent. This class is particularly relevant in the theory of aperiodic order, with several applications to mathematical quasicrystals and Meyer sets; see [4,8,11,12,14,17,18] and references therein.…”

Uniqueness theorem for Fourier transformable measures on LCA groups

“…, which contradicts to (9) for sufficiently small ε. As above, take the sets V, V k and the functions ϕ j (x), Ψ(x) satisfying conditions (7), (8), and (9). Fix b ∈ G, suppose that…”

“…Hereĝ is the Fourier transform of g, andǧ is the inverse Fourier transform. Fourier transformable discrete measures were considered in a series of papers [5]- [9]. A function g ∈ C u (G) is almost periodic if the closure of the family of translates {g(• − t)} t∈G is a compact subset of C u (G).…”

Diffraction of fully Euclidean model sets

“…Studies regarding the behaviour of model sets were initially restricted to the fully Euclidean case; the setting where everything lives in some real space. Today the theory has advanced to the more abstract case of locally compact Abelian groups, see [2,5,26,27,35,36] for some examples. It is for simplicity that we restrict ourselves to the fully Euclidean case.…”