Poisson Summation Formulas for Cut and Project Quasicrystals In Lagarias' paper "Mathematical Quasicrystals and the Problem of Diffraction" http://www.math.lsa.umich.edu/~lagarias/doc/diffraction.pdf he discusses various ways one might get Poisson summation formulas for certain nonuniform point sets in the plane. More precisely, given a Delone set $\Lambda \subset \mathbb{R}^n$ obtained using a cut and project scheme, consider the tempered distribution
$$\mu_\Lambda = \sum_{x \in \Lambda} \delta_x.$$
We would like to compute $\hat{\mu}_\Lambda$. In section 3 (specifically Thm 3.9 and the surrounding discussion) he seems to indicate that for cut and project sets, we know something about $\hat{\mu}_\Lambda$, but I can't tell how to interpret Thm 3.9 in that context. 
Question 1: How close does Thm 3.9 get us to computing $\hat{\mu}_\Lambda$?
Question 2: Do we at least know that $\hat{\mu}_\Lambda$ is a tempered distribution of the form $$\sum_{x \in \Lambda'} c_x\delta_x$$
for some Delone set $\Lambda'$? If so, how much do we know about the set $\Lambda'$?
Edit: In Meyer's paper "Quasicrystals, diophantine approximation and algebraic numbers" he computes a variety of weighted Poisson summation formulas, but falls short of a computation of $\hat{\mu}_\Lambda$. These computations appear to have been done in the 70's. Has anyone tried to improve his results since then?
 A: Question 1 let $\gamma$ be the auto-correlation of $\Lambda$. 
It can be proven that
$$ \lim_n \frac{\int_{x+A_n} \bar{\chi(t)} d \mu_\Lambda (t)}{vol(A_n)}(*)$$
exists uniformly in $x$. In the case of Fourier Transformable measures, this limit is exactly $\widehat{\mu_\Lambda}(\{ \chi \})$.  
It follows then from a result of Hof that
$$ \widehat{\gamma}(\{ \chi \}) = \left| \lim_n \frac{\int_{A_n} \bar{\chi(t)} d \mu_\Lambda (t)}{vol(A_n)} \right|^2 \,.$$
I think the existence of the limit (*) is covered in a paper by Robert Moody (1).
Also the problem of when the two limits can be connected without the uniformity in $x$ (aka the Bombieri Taylor conjecture) is covered pretty well in a paper by Daniel Lenz (2) available here
Question 2 The answer is no in general. More precisely, the answer is yes if and only if $\Lambda=L+F$ where $L$ is lattice and $F$ is finite.
There is an old paper by Cordoba (3) which shows that this is not the case, and Lagarias mentions this paper. The proof is pretty technical, and I heard that there might be some issues with the proof though.
In this case, since $\Lambda$ is a Meyer set, there exists a much simpler proof of this claim, which can be found in my paper (Proposition 7.3 in (4))archiv link
A much stronger version of this claim was proven recently by Favorov (5) and independently by Kellendonk and Lenz (6) They basically proved that if $\Lambda$ is a Delone set with FLC, and $\delta_\Lambda$ is Fourier Transformable, with a discrete FT, then  $\Lambda=L+F$ where $L$ is lattice and $F$ is finite.
I am actually pretty sure that in general, $\widehat{\mu_\Lambda}$ cannot even be a measure, you have to always treat it as a tempered distribution.
Bibliography


*

*R. V. Moody,  Uniform distribution in model sets,  Canad. Math. Bull. Vol. 45 (1), 123-130, 2002.

*D. Lenz, Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks, Commun. Math. Phys. 287 , 225-258, 2009.

*A. Cordoba,  Dirac combs, Lett. Math. Phys., 17, 191-196, 1989.

*N.Strungaru,  On the Bragg Diffraction Spectra of a Meyer Set, Canadian Journal of Mathematics 65, no. 3, 675-701, 2013.

*Favorov  Bohr and Besicovitch almost periodic discrete sets and quasicrystals, Proc. Amer. Math. Soc. 140, 1761-1767, 2012.

*J. Kellendonk, D. Lenz,  Equicontinuous delone dynamical systems , Canadian Journal of Mathematics 65, 149--170, 2013.

