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Let $f=\sum_{\lambda\in\Lambda}\sum_k p_k(\lambda) D^k\delta_\lambda$ be a temperate distribution on $\mathbb{R}^d$ with uniformly discrete support $\Lambda$ and uniformly discrete spectrum (that is $supp\hat f$). We prove that under conditions
$$
0 < c\le\sum_k|p_k(\lambda)|\le C < \infty
$$
the support $\Lambda$ is a finite union of cosets of full-rank lattices. The optimality of the above estimates is discussed. The result generalizes the corresponding one for discrete measures [1]. For its proof we use some properties of almost periodic distributions and a local version of Wiener's Theorem on trigonometric series.
[1] S.Yu.Favorov, Large Fourier Quasicrystals and Wiener's Theorem, Journal of Fourier Analysis and Applications, Vol. 25, Issue 2, (2019), 377-392.
