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

##### Author
Baake, Michael
Strungaru, Nicolae
Terauds, Venta
2020
##### Keywords
Abelian groups , compact groups , measure theory , Radon transforms , weights & measures , Radon
##### Abstract (summary)
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.
##### Publication Information
Baake, M., Strungaru, N., & Terauds, V. (2020). Pure point measures with sparse support and sparse Fourier-Bohr support. Transactions of the London Mathematical Society, 7(1), 1–32. https://doi.org/10.1112/tlm3.12020
Article
English