# Pure point measures with sparse support and sparse Fourier--Bohr support

## TL;DR

This paper extends the theory of doubly sparse Fourier-transformable measures from Euclidean spaces to locally compact Abelian groups, characterizing their structure and spectrum using cut and project schemes and almost periodic measures.

## Contribution

It generalizes the understanding of doubly sparse measures to more abstract groups, providing new characterizations and representations in this broader setting.

## Key findings

- Characterization of sparseness of Fourier--Bohr spectrum for measures with Meyer set support
- Representation of measures in terms of trigonometric polynomials
- Derivation of a Poisson summation type formula for doubly sparse measures

## 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.