Exercise 59, § 1

Let $n\geqslant 1$ be an integer and equip $\mathbf{R}^n$ with the euclidean norm. Let $\mu$ denote Lebesgue measure on $\mathbf{R}^n$.

Let $r >0$. A sphere packing of radius $r$ in $\mathbf{R}^n$ is a set $\mathscr{P}$ of disjoint parts of $\mathbf{R}^n$, such that every $S\in \mathscr{P}$ is a closed ball of radius $r$. One says that $\mathscr{P}$ is periodic if there exists a lattice $\Lambda \subset \mathbf{R}^n$ such that $S +h\in \mathscr{P}$ for every $S\in \mathscr{P}$ and every $h\in \Lambda$.

Let $A_{\mathscr{P}}$ be the union of the elements of $\mathscr{P}$. It is a measurable set. One calls the density of $\mathscr{P}$ the real number

$\Delta (\mathscr{P}) =$ lim sup$_{R\rightarrow+\infty}\frac{\mu(A_{\mathscr{P}}\cap B_R)}{\mu(B_R)}$ where $B_R$ is the closed ball of radius R in $\mathbf{R}^n$. Let $\Delta_n=$ sup$_{\mathscr{P}}\Delta (\mathscr{P})$. (a) One has

$\Delta_n=$ sup $\Delta (\mathscr{P})$.

$\mathscr{P}$ periodic

(b) Let $\mathscr{P}$ be a periodic sphere packing, invariant under the lattice Λ. Let $C\subset \mathbf{R}^n$ be a set of representatives of the centres of the balls $S\in \mathscr{P}$ modulo Λ. One has

$\Delta (\mathscr{P}) =\mu(B_1)r^n$Card(C)V(Λ)

where V(Λ) is the covolume of Λ.

(c) Let $a >0$ be a real number. Let $f:\mathbf{R}^n\rightarrow \mathbf{R}$ be a Schwartz function such that

(i) One has $f(x)\leqslant 0$ if $\|x\|\geqslant 2a$;

(ii) The Fourier transform of $f$ has positive real values and satisfies $\widehat{f}(0)>0$.

Then one has

(It suffices to consider periodic sphere packings of radius $a$; let $\mathscr{P}$ be one of them, and Λ, $C\subset \mathbf{R}^n$ as in the preceding question; using the Poisson formula, prove that

$(c,d)\in C^2n\in \Lambda h\in \Lambda^*c\in C$

where $\Lambda^*$ is the associated lattice, and conclude.)