### Remarks on sphere packings, clusters and Hales Ferguson theorem

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Let $\beta \>1$ be an algebraic number. We study the strings of zeros (“gaps”) in the Rényi $\beta $-expansion ${d}_{\beta}\left(1\right)$ of unity which controls the set ${\mathbb{Z}}_{\beta}$ of $\beta $-integers. Using a version of Liouville’s inequality which extends Mahler’s and Güting’s approximation theorems, the strings of zeros in ${d}_{\beta}\left(1\right)$ are shown to exhibit a “gappiness” asymptotically bounded above by $log\left(\mathrm{M}\right(\beta \left)\right)/log\left(\beta \right)$, where $\mathrm{M}\left(\beta \right)$ is the Mahler measure of $\beta $. The proof of this result provides in a natural way a new classification of algebraic numbers $\>1$ with classes called Q...

The set $\mathcal{U}{\mathcal{D}}_{r}$ of point sets of ${\mathbb{R}}^{n},n\ge 1$, having the property that their minimal interpoint distance is greater than a given strictly positive constant $r\>0$ is shown to be equippable by a metric for which it is a compact topological space and such that the Hausdorff metric on the subset $\mathcal{U}{\mathcal{D}}_{r,f}\subset \mathcal{U}{\mathcal{D}}_{r}$ of the finite point sets is compatible with the restriction of this topology to $\mathcal{U}{\mathcal{D}}_{r,f}$. We show that its subsets of Delone sets of given constants in ${\mathbb{R}}^{n},n\ge 1$, are compact. Three (classes of) metrics, whose one of crystallographic nature,...

We investigate in a geometrical way the point sets of $\mathbb{R}$ obtained by the $\beta $-numeration that are the $\beta $-integers ${\mathbb{Z}}_{\beta}\subset \mathbb{Z}\left[\beta \right]$ where $\beta $ is a Perron number. We show that there exist two canonical cut-and-project schemes associated with the $\beta $-numeration, allowing to lift up the $\beta $-integers to some points of the lattice ${\mathbb{Z}}^{m}$ ($m=$ degree of $\beta $) lying about the dominant eigenspace of the companion matrix of $\beta $ . When $\beta $ is in particular a Pisot number, this framework gives another proof of the fact that ${\mathbb{Z}}_{\beta}$ is...

The Fourier transform of a weighted Dirac comb of beta-integers is characterized within the framework of the theory of Distributions, in particular its pure point part which corresponds to the Bragg part of the diffraction spectrum. The corresponding intensity function on this Bragg part is computed. We deduce the diffraction spectrum of weighted Delone sets on beta-lattices in the split case for the weight, when beta is the golden mean.

The spectrum of a weighted Dirac comb on the Thue-Morse quasicrystal is investigated by means of the Bombieri-Taylor conjecture, for Bragg peaks, and of a new conjecture that we call Aubry-Godrèche-Luck conjecture, for the singular continuous component. The decomposition of the Fourier transform of the weighted Dirac comb is obtained in terms of tempered distributions. We show that the asymptotic arithmetics of the $p$-rarefied sums of the Thue-Morse sequence (Dumont; Goldstein, Kelly and Speer; Grabner;...

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