Why do (weak) Meyer sets diffract?

Tempered distributions with translation bounded measure as Fourier transform and the generalized Eberlein decomposition

“…Proof (a)This follows from [34, Theorem 4.1] or [36, Theorem 5.7]. (b)This is a consequence of [29, Corollary 35]. (c)By Corollary 5.11, we have $$f \in \mathcal {WLS}_2({\mathbb {R}}^d)$$. The claim follows now from Theorem 5.13.$$\Box$$ …”

“…Remark Under the assumptions of Theorem 5.15, if $$({\mathbb {R}}^d, H, {\mathcal {L}})$$ is any cut‐and‐project scheme and $$W \subseteq H$$ any compact set such that $$\Lambda \subseteq \mbox{$\curlywedge $}(W)$$, we can choose $$\Gamma = \mbox{$\curlywedge $}(W)$$ [36, Theorem 5.7].…”

“…The generalized Eberlein decomposition exists for Fourier transformable measures with Meyer set support [34, 36]. This result has important consequences for one‐dimensional Pisot substitutions, see, for example, [2, 4], and we expect that establishing the existence of the generalized Eberlein decomposition may have important consequences in general.…”

“…\end{equation*}$$This allows us to study the pure point diffraction spectrum $$(\widehat{\gamma })_{\operatorname{pp}}$$ and continuous diffraction spectrum $$(\widehat{\gamma })_{\operatorname{c}}$$ of μ, respectively, in the Fourier dual space by studying instead the components $$\gamma _{\operatorname{s}}$$ and γ 0 , respectively, of the autocorrelation measure γ in the real space. This approach has proven effective in the study of diffraction of measures with Meyer set support [31–34, 36] and in the study of compatible random substitutions [7]. Because of this, one would like to further decompose $$$$\begin{equation*} \gamma _{0}= \gamma _{0a}+\gamma _{0s}, \end{equation*}$$$$such that Whenever this is possible, we will refer to $$$$\begin{equation} \gamma =\gamma _{\operatorname{s}}+\gamma _{0a}+\gamma _{0s} \end{equation}$$$$as the complete (or generalized) Eberlein decomposition of γ.…”