2.1 Bounding inhomogeneous Gaussian lattice sums: Far away from deep holes

We revisit some of the arguments originally due to Bétermin and Petrache [Reference Bétermin and Petrache7, Theorem 1.5], in order to obtain the following result that meets our needs:

Theorem 2.1. Let $c\in V(L)$ be a deep hole of L, and let $\varrho $ such that $\varrho < \mu (L)$ . Then there exists $\alpha _\varrho $ such that for any $\alpha> \alpha _\varrho $ , for every z in the ball $B(0, \varrho )$ ,

$$\begin{align*}p(f_\alpha, L, z)> p(f_\alpha, L, c). \end{align*}$$

This result is an immediate consequence of the following inequality:

Lemma 2.2. Let L be an n-dimensional lattice, let c be a deep hole of L, and let $z \in \mathbb {R}^n$ be an arbitary point. If $\alpha> n / (2 \mu (L)^2)$ , then

(2) $$ \begin{align} \left(\sum_{x \in L} e^{-\alpha \|x-c\|^2} \right) \bigg/ \left(\sum_{x \in L} e^{-\alpha \|x-z\|^2} \right) \leq e^{-\alpha(\mu(L)^2 - \|z\|^2)} \left(\frac{2\alpha \mu(L)^2 e}{n} \right)^{n/2}. \end{align} $$

Indeed, for every z in $B(0, \varrho )$ ,

$$\begin{align*}\mu(L)^2 - \|z\|^2> \mu(L)^2 - \varrho^2 > 0, \end{align*}$$

and

$$\begin{align*}e^{-\alpha(\mu(L)^2 - \varrho^2)} \left(\frac{2\alpha\mu(L)^2 e}{n} \right)^{n/2} \end{align*}$$

goes to $0$ when $\alpha $ tends to infinity. Note that this proof gives an explicit estimate on $\alpha _\varrho $ depending on $\varrho $ .

It remains to prove Lemma 2.2. For this, we need estimates by Banaszczyk [Reference Banaszczyk3], which in the following variant can be found in [Reference Barvinok5, Lemma 18.2 and Problem 18.4.1].

Proof of Lemma 2.2.

We can write

$$\begin{align*}\sum_{x \in L} e^{-\alpha\| x - c\|^2} = \sum_{x \in \sqrt{\alpha/\pi} L} e^{-\pi \| x - \sqrt{\alpha/\pi} c\|^2}. \end{align*}$$

Now $\sqrt {\alpha /\pi } c$ is a deep hole of the scaled lattice $\sqrt {\alpha /\pi } L$ . For a given radius r, we can split the last sum into two parts,

$$\begin{align*}\sum_{x \in \sqrt{\alpha/\pi } L, \| x - \sqrt{\alpha/\pi} c\| \leq r } e^{-\pi \| x - \sqrt{\alpha/\pi} c\|^2} + \sum_{x \in \sqrt{\alpha/\pi} L, \| x - \sqrt{\alpha/\pi} c\|> r } e^{-\pi \| x - \sqrt{\alpha/\pi} c\|^2}. \end{align*}$$

Observe that the first sum vanishes whenever r is strictly smaller than $\sqrt {\alpha /\pi } \mu (L)$ , the covering radius of $\sqrt {\alpha /\pi } L$ . If $\alpha $ is large enough, we can choose r in the range $\sqrt {\frac {n}{2\pi }} < r < \sqrt {\alpha /\pi } \mu (L)$ so that we can apply (a) of Lemma 2.3. This gives

$$\begin{align*}\begin{aligned} \sum_{x \in L} e^{-\alpha \|x-c\|^2} & = \sum_{x \in \sqrt{\alpha/\pi} L, \| x - \sqrt{\alpha/\pi} c\|> r } e^{-\pi \| x - \sqrt{\alpha/\pi} c\|^2}\\ & \leq e^{-\pi r^2} \left(\frac{2 \pi e r^2}{n} \right)^{n/2} \sum_{x \in \sqrt{\alpha/\pi} L} e^{-\pi\|x\|^2}. \end{aligned} \end{align*}$$

Now we apply (b) of Lemma 2.3,

$$\begin{align*}\sum_{x \in L} e^{-\alpha \|x-c\|^2} \leq e^{-\pi r^2} \left(\frac{2 \pi e r^2}{n} \right)^{n/2} e^{-\alpha\|z\|^2} \sum_{x \in L} e^{-\alpha\|x-z\|^2}. \end{align*}$$

By letting r tending to $\sqrt {\alpha /\pi } \mu (L)$ we get the desired inequality (2).

Based on Lemma 2.3, Bétermin and Petrache obtain the following statement [Reference Bétermin and Petrache7, Theorem 1.5], which is an immediate consequence of Theorem 2.1.

Corollary 2.4. Let L be an n-dimensional lattice and let c be a deep hole of L. Then for any $z\in \mathbb {R}^n$ , which is not a deep holeFootnote ¹, there exists $\alpha _z$ such that for any $\alpha> \alpha _z$ ,

$$\begin{align*}p(f_\alpha, L, z)> p(f_\alpha, L, c). \end{align*}$$

Thus, for $\alpha \to \infty $ cold spots tend to deep holes.

Note that the arguments given above are not strong enough to prove that cold spots are deep holes for finite $\alpha $ . The quantity $\mu (L)^2 - \| z \|^2$ cannot be uniformly bounded from below with respect to z in the Voronoi cell of L.

2.2 Bounding inhomogeneous Gaussian lattice sums: Using the linear programming bound for spherical designs around deep holes

Here we prove that if every shell around a deep hole is a spherical $2$ -design, then the deep hole is a (strict) local minimizer when $\alpha $ is large enough. That is, we show:

Moreover, explicit estimates on $\alpha _c$ and $R_{\alpha _c}$ will be given in Lemma 2.9.

Qualitatively, it is easy to see that c is a local minimizer for $\alpha $ large enough, using the computation of the Hessian in (1), which for the Gaussian core model evaluates to

$$\begin{align*}\nabla^2 p(f_\alpha, L, c) = \sum_{r \geq 0} |L(c,r)| \left( \frac{4\alpha^2 r^2}{n} - 2\alpha\right) e^{-\alpha r^2} I_n. \end{align*}$$

Indeed, if $\alpha $ is sufficiently large, then all summands are positive multiples of the identity matrix. This ensures the existence of a radius $R_\alpha>0$ such that c is a minimizer in the ball $B(c,R_\alpha )$ .

However, as we aim to combine this local information with Theorem 2.1 to obtain a global bound, we need to derive a uniform lower bound for $R_\alpha $ when $\alpha $ is large enough. The formula for the Hessian is easy to evaluate at a point z only when all inhomogeneous shells around z are spherical $2$ -designs. So information about the Hessian at points close to c seem to be difficult to get. A more delicate analysis is required.

For this, we will heavily use the linear programming bound for spherical designs, originally developed by Fazekas and Levenshtein [Reference Fazekas and Levenshtein20] and Yudin [Reference Yudin27] to determine lower bounds for the covering radius of spherical designs. Later, in the context of polarization problems on spheres, this linear programming bound was modified in [Reference Boyvalenkov, Dragnev, Hardin, Saff and Stoyanova12] and [Reference Borodachov10].

Let us recall the details of this linear programming bound. Let $a : [-1,1] \to \mathbb {R}$ be any function. We are interested in determining

$$\begin{align*}E_a(X) = \min\left\{\sum_{x \in X} a(x \cdot y) : y \in S^{n-1}\right\}. \end{align*}$$

To state the linear programming bound for $E_a$ we need to introduce specific polynomials $P^n_k$ . They are univariate polynomials of degree k, normalized by $P^n_k(1) = 1$ , and orthogonal in the following sense

$$\begin{align*}\int_{-1}^1 P^n_k(t) P^n_l(t) (1-t^2)^{(n-3)/2} dt = 0 \quad \text{ if } k \neq l. \end{align*}$$

In fact, the polynomials $P^n_k$ are appropriate multiples of the Jacobi polynomials $P^{(n-3)/2,(n-3)/2}_k$ , see for instance the book [Reference Andrews, Askey and Roy1] by Andrews, Askey, and Roy. Note that we have

$$\begin{align*}P_0^n(t) = 1, \quad P_1^n(t) = t,\quad P_2^n(t) = \frac{1}{n-1}(nt^2-1). \end{align*}$$

Proof. The inequality follows easily from orthogonality and the spherical design property. For $y \in S^{n-1}$ ,

$$\begin{align*}\begin{aligned} E_a(X) & \geq \sum_{x \in X} a(x \cdot y) \geq \sum_{x \in X} h(x \cdot y) = |X| \frac{1}{|X|} \sum_{x \in X} \sum_{k=0}^M h_k P_k^n(x \cdot y)\\ & = |X| \sum_{k= 0}^M h_k \int_{S^{n-1}} P_k(x \cdot y) \, d\omega(x)\\ & = |X| \sum_{k= 0}^M h_k \int_{-1}^1 P_k(t) (1-t^2)^{(n-3)/2} \, dt\\ & = |X| h_0. \end{aligned}\\[-38pt] \end{align*}$$

As usual, the proof of the previous lemma provides additional information about sharp cases; that is, when there is a polynomial h and a point $y \in S^{n-1}$ with

$$\begin{align*}h_0 |X| = E_a(X) = \sum_{x \in X} a(x \cdot y). \end{align*}$$

Indeed, one sees from the proof that this can only happen when $a(x \cdot y) = h(x \cdot y)$ for $x \in X$ . Since we also need $h(t) \leq a(t)$ for all $t \in [-1,1]$ this also means that, if a is differentiable, h and a need to coincide up to second order when $x \cdot y \in (-1,1)$ for $x \in X$ .

Figure 3 Our strategy is to bound the inhomogeneous Gaussian lattice sum close to the deep hole c using the linear programming bound for every inhomogeneous shell $L(c,r)$ and for the points on every concentric sphere $c + \rho y$ , with $y \in S^{n-1}$ , around the deep hole c.

To bound $p(f_\alpha , L, z)$ for points z close to a deep hole c we apply the linear programming bound for every concentric sphere $z = c + \rho y$ , with $\rho> 0$ and $y \in S^{n-1}$ , and for every inhomogeneous shell $L(c,r)$ ;

$$\begin{align*}\begin{aligned} p(L,f_\alpha, z) & = \sum_{x \in L} e^{-\alpha \|x - z\|^2} = \sum_{x \in L} e^{-\alpha\|x - c - \rho y\|^2} \\ & = \sum_{r \geq 0} e^{-\alpha r^2} e^{-\alpha \rho^2} \sum_{x \in L(c,r)} e^{2\alpha \rho r ((x-c)/r) \cdot y}; \end{aligned} \end{align*}$$

see Figure 3.

Lemma 2.7. Suppose the inhomogeneous shell $L(c,r)$ is a spherical $2$ -design. Then

(3) $$ \begin{align} \sum_{x \in L(c,r)} e^{2\alpha \rho r ((x-c)/r) \cdot y} \geq |L(c,r)| b(\alpha, \rho, r) \end{align} $$

holds with

$$\begin{align*}b(\alpha, \rho, r) = \frac{ne^{\frac{2\alpha \rho r}{n}}+e^{-2\alpha \rho r}}{n+1}, \end{align*}$$

and if $L(c,r)$ is additionally centrally symmetric, then

$$\begin{align*}b(\alpha, \rho, r) = \cosh\left(\frac{2\alpha \rho r}{\sqrt{n}}\right). \end{align*}$$

Remark 2.8. The bounds of the previous lemma are in fact optimal.

In the first case, the bound is tight, when X are the vertices of a regular simplex and when $y = - v$ , where v is any vertex of the simplex. Then the inner products between y and the elements of X are the interpolation points $-1$ and $1/n$ . Therefore,

$$\begin{align*}E_{a_\gamma}(X) \leq \sum_{x\in X} a_\gamma (x \cdot y) = \sum_{x\in X} h_\gamma (x \cdot y) = h_0 |X| \leq E_{a_\gamma}(X), \end{align*}$$

which shows the optimality of the bound.

In the second case, the bound is tight when X are the vertices of a cross polytope, namely $X=\{\pm e_i\}$ , where $e_i$ is the i-th vector of the canonical basis. For $y = \frac {1}{\sqrt {n}} \sum _{i = 1}^n e_i$ , then the inner products between y and the elements of X are the interpolation points $\pm \frac {1}{\sqrt {n}}$ , which proves optimality.

With Lemma 2.7, we have

$$\begin{align*}p(L,f_\alpha, z) \geq \sum_{r \geq 0} |L(c,r)| e^{-\alpha r^2} e^{-\alpha \rho^2} b(\alpha, \rho, r). \end{align*}$$

On the other hand,

$$\begin{align*}p(f_\alpha, L, c) = \sum_{r \geq 0} |L(c,r)| e^{-\alpha r^2}. \end{align*}$$

Thus, if for a given $\alpha $ , we find R such that for every $0< \rho < R$ , and for every r the strict inequality

(4) $$ \begin{align} e^{-\alpha \rho^2} b(\alpha,\rho,r)> 1 \end{align} $$

holds, then we get that for every $z\neq c$ in the ball $B(c,R)$ ,

$$\begin{align*}p(f_\alpha, L, z) - p(f_\alpha, L, c) \geq \sum_{r \geq 0} |L(c,r)| e^{-\alpha r^2} (e^{-\alpha \rho^2} b(\alpha,\rho,r) - 1)> 0. \end{align*}$$

We need to verify inequality (4). Note that since both expressions $b(\alpha , \rho , r)$ are increasing with r, if (4) is satisfied for $r = \mu (L)$ , then it is satisfied for every $r \geq \mu (L)$ .

We therefore consider the function

$$\begin{align*}g(\rho) = e^{-\alpha \rho^2}b(\alpha,\rho,\mu(L)) - 1, \end{align*}$$

and look for $R_\alpha>0$ such that for every $0 < \rho < R_\alpha $ , $g(\rho )>0$ .

Lemma 2.9. In the general case, when the inhomogeneous shells are spherical 2-designs, then for every $\alpha $ such that $\alpha> \frac {n^2}{ \mu (L)^2}(\frac {n+1}{2n} + \log (\frac {n+1}{n})^2)$ , then $g(\rho )> 0$ on the interval $\left (0, R_\alpha \right ]$ , where

$$\begin{align*}R_\alpha = \frac{\mu(L)}{n} + \sqrt{\frac{\mu(L)^2}{n^2}- \frac{\log(\frac{n+1}{n})}{\alpha}}. \end{align*}$$

When, moreover, the inhomogeneous shells are centrally symmetric, then the inequality $\alpha> \frac {n}{ \mu (L)^2}(1/2 + \log (2)^2)$ is sufficient, and we can take

$$\begin{align*}R_\alpha = \frac{\mu(L)}{\sqrt{n}} + \sqrt{\frac{\mu(L)^2}{n}- \frac{\log(2)}{\alpha}}. \end{align*}$$

Proof. We focus on the general case where

$$\begin{align*}g(\rho) = e^{-\alpha \rho^2}\left(\frac{ne^{\frac{2\alpha \rho \mu(L)}{n}}+e^{-2\alpha \rho \mu(L)}}{n+1}\right) - 1, \end{align*}$$

because the second, more specific case, is similar. The strategy consists in using two different estimates: a first one to ensure the positivity of $g(\rho )$ for $\rho $ in an interval of the form $[r_\alpha , R_\alpha ]$ , and second one, using the Taylor expansion of g around $0$ , to make sure that $g(\rho )>0$ for $\rho \in (0,r_\alpha )$ .

First, we use the trivial lower bound

$$\begin{align*}\frac{ne^{\frac{2\alpha \rho \mu(L)}{n}}+e^{-2\alpha \rho \mu(L)}}{n+1}> \frac{ne^{\frac{2\alpha \mu(L) \rho}{{n}}}}{n+1}, \end{align*}$$

which gives

$$\begin{align*}g(\rho)> \frac{ne^{\alpha \rho\left(\frac{2\mu(L)}{n}-\rho\right)}-(n+1)}{n+1} \end{align*}$$

and therefore $g(\rho )>0$ whenever

$$\begin{align*}\alpha \rho\left(\frac{2\mu(L)}{n}-\rho\right)> \log\left(\frac{n+1}{n}\right). \end{align*}$$

Because $\frac {n+1}{2n} + \log \left (\frac {n+1}{n}\right )^2> \log \left (\frac {n+1}{n}\right )$ , the assumption on $\alpha $ ensures that $g(\rho )>0$ on the interval

$$\begin{align*}\left[ \frac{\mu(L)}{{n}} - \sqrt{\frac{\mu(L)^2}{n^2} - \frac{\log\left(\frac{n+1}{n}\right)}{\alpha}}, \frac{\mu(L)}{{n}} + \sqrt{\frac{\mu(L)^2}{n^2} - \frac{\log\left(\frac{n+1}{n}\right)}{\alpha}} \right] .\end{align*}$$

It remains to deal with small $\rho $ . Note that because $\sqrt {1-x} \geq 1-x$ for every $0\leq x \leq 1$ , we have

$$\begin{align*}\frac{\mu(L)}{n} - \sqrt{\frac{\mu(L)^2}{n^2} - \frac{\log\left(\frac{n+1}{n}\right)}{\alpha}} = \frac{\mu(L)}{n} - \frac{\mu(L)}{n}\sqrt{1 - \frac{\log\left(\frac{n+1}{n}\right)n^2}{\alpha\mu(L)^2}} \leq \frac{\log\left(\frac{n+1}{n}\right){n}}{\alpha\mu(L)} \end{align*}$$

and therefore it is sufficient to prove that $g(\rho )>0$ on $\left (0,\frac {\log \left (\frac {n+1}{n}\right ){n}}{\alpha \mu (L)} \right ]$ . For $\rho $ in this interval, we have

$$\begin{align*}e^{-\alpha \rho^2} \geq 1 -\alpha \rho^2> 0. \end{align*}$$

Moreover, since

$$\begin{align*}e^{\frac{2\alpha \rho \mu(L)}{n}}> 1 + \frac{2\alpha \rho \mu(L)}{n} + \frac{2\alpha^2 \rho^2 \mu(L)^2}{n^2} \end{align*}$$

and

$$\begin{align*}e^{-2\alpha \rho \mu(L)}> 1 - 2\alpha \rho \mu(L), \end{align*}$$

then

$$\begin{align*}ne^{\frac{2\alpha \rho \mu(L)}{n}}+e^{-2\alpha \rho \mu(L)}> n+1 + \frac{2\alpha^2 \rho^2 \mu(L)^2}{n}. \end{align*}$$

We get

$$\begin{align*}g(\rho)> (1 -\alpha \rho^2)\left(1 + \frac{2\alpha^2 \mu(L)^2 \rho^2}{n(n+1)}\right) - 1 = -\alpha \rho^2\left( 1 - \frac{2\alpha \mu(L)^2}{n(n+1)} + \frac{2\alpha^2 \mu(L)^2 \rho^2}{n(n+1)} \right), \end{align*}$$

which is positive between $0$ and

$$\begin{align*}\sqrt{\frac{1}{\alpha} - \frac{n(n+1)}{2\alpha^2 \mu(L)^2}} = \sqrt{\frac{\alpha - \frac{n(n+1)}{2\mu(L)^2}}{\alpha^2}}. \end{align*}$$

Thus $g(\rho )> 0$ on $\left (0,\frac {\log \left (\frac {n+1}{n}\right ){n}}{\alpha \mu (L)} \right ]$ whenever $\alpha - \frac {n(n+1)}{2\mu (L)^2}> \frac {\log \left (\frac {n+1}{n}\right )^2n^2}{\mu (L)^2}$ , which is ensured by the assumption on $\alpha $ .

For the second case, we use the similar estimates

$$\begin{align*}\cosh\left(\frac{2\alpha \mu(L) \rho}{\sqrt{n}}\right)> \frac{e^{\frac{2\alpha \mu(L) r}{\sqrt{n}}}}{2}, \end{align*}$$

and

$$\begin{align*}g(\rho)> (1 -\alpha \rho^2)\left(1 + \frac{2\alpha^2 \mu(L)^2 \rho^2}{n}\right) - 1 = -\alpha \rho^2\left( 1 - \frac{2\alpha \mu(L)^2}{n} + \frac{2\alpha^2 \mu(L)^2 \rho^2}{n} \right).\\[-34pt] \end{align*}$$

From Lemma 2.9, we can easily deduce Theorem 2.5:

2.3 Putting the two bounds together, finishing the proof of Theorem 1.2

The assumptions of Theorem 1.2 allow us to apply Theorem 2.5. We have for every $\alpha \geq \alpha _c$ and for every $z \neq c$ in the ball $B(c,R_{\alpha _c})$ ,

$$\begin{align*}p(f_\alpha, L, z)> p(f_\alpha, L, c). \end{align*}$$

Since the deep holes of L are precisely the points in $V(L)$ having norm $\mu (L)$ , there exists $0 < \varrho < \mu (L)$ such that

$$\begin{align*}V(L) \setminus \bigcup_{c} B(c, R_{\alpha_c}) \subseteq B(0, \varrho), \end{align*}$$

where the union is taken over all the deep holes in $V(L)$ . Figure 4 illustrates this covering argument.

Figure 4 Covering $V(L)$ by balls of two kinds.

We can now apply Theorem 2.1, which provides $\alpha _\varrho $ such that for every $\alpha> \alpha _\varrho $ and every $z \in B(0, \varrho )$ ,

$$\begin{align*}p(f_\alpha, L, z)> p(f_\alpha, L, c). \end{align*}$$

The desired result then follows by taking $\alpha _0 = \max \{\alpha _c, \alpha _\varrho \}$ .

3.1 Decomposable lattices

We will call a lattice L in $\mathbb {R}^n$ decomposable if we can find non trivial sublattices $L_1$ on $V_1 = \mathbb {R} L_1$ and $L_2$ on $V_2 = \mathbb {R} L_2$ with

$$\begin{align*}L = L_1 \perp L_2, \end{align*}$$

otherwise L is called indecomposable. A general feature of decomposable lattices is that (inhomogeneous) energy factors multiplicatively through the orthogonal summands.

Lemma 3.1. Let $L\subset \mathbb {R}^n$ be a lattice that can be decomposed as the orthogonal direct sum of lattices

$$\begin{align*}L = L_1 \perp L_2 \perp \ldots \perp L_m, \end{align*}$$

namely there are subspaces $V_1, \ldots , V_m \subset \mathbb {R}^n$ such that $L_i$ is a lattice in $V_i$ for every $1 \leq i \leq m$ , and $\mathbb {R}^n = V_1 \perp V_2 \perp \ldots \perp V_m$ . Then for any $\alpha> 0$ and $z = z_1 + \ldots + z_m$ with $z_i \in V_i$ , we have

$$\begin{align*}p(f_\alpha, L, z) = p(f_\alpha, L_1, z_1)p(f_\alpha, L_2, z_2) \cdots p(f_\alpha, L_m, z_m). \end{align*}$$

In particular, if for every $1 \leq i \leq m$ , $c_i$ is a minimizer of $z_i \mapsto p(f_\alpha , L_i, z_i)$ , then $c = c_1 + \ldots + c_m$ is a minimizer of $z \mapsto p(f_\alpha , L, z)$ .

Proof. It is enough to prove the statement when $L = L_1 \perp L_2$ . Then, for $z = z_1 + z_2$ , we have

$$ \begin{align*} p(f_\alpha, L, z) & = \sum_{\ell \in L}e^{-\pi\alpha \| \ell - z\|^2} \\ & = \sum_{\ell_1 \in L_1, \ell_2 \in L_2}e^{-\pi\alpha \| \ell_1 + \ell_2 - z_1 - z_2\|^2} \\ & = \sum_{\ell_1 \in L_1, \ell_2 \in L_2}e^{-\pi\alpha \| \ell_1 - z_1 \|^2}e^{-\pi\alpha \| \ell_2 - z_2\|^2} \\ & = \left(\sum_{\ell_1 \in L_1}e^{-\pi\alpha \| \ell_1 - z_1 \|^2}\right)\left(\sum_{\ell_2 \in L_2}e^{-\pi\alpha \| \ell_2 - z_2\|^2}\right) \\ & = p(f_\alpha, L_1, z_1)p(f_\alpha, L_2, z_2).\\[-34pt] \end{align*} $$

3.2 Designs on inhomogeneous shells of indecomposable root lattices

An integral lattice L is a root lattice if and only if

(5) $$ \begin{align} L = \langle L(2) \rangle_{\mathbb{Z}}, \end{align} $$

that is, if L is generated by its elements of squared norm $2$ , which in fact is a root system. Note that as a direct consequence of the definition all root lattices are even, that is $\|v\|^2 \in 2\mathbb {Z}$ for all $v \in L$ .

If L is decomposable, say $L = L_1 \perp \ldots \perp L_m$ with $L_1,\ldots ,L_m$ indecomposable, then this is true for the root system

$$\begin{align*}L(2) = L_1(2) \perp \ldots \perp L_m(2). \end{align*}$$

It is well known that in this case each root system $L_i(2)$ is indecomposableFootnote ² as a root system and is of type $A_n$ ( $n \geq 2$ ), $D_n$ ( $n \geq 3$ ) or $E_n$ ( $n \in \left \{ 6,7,8 \right \}$ ) (c.f. [Reference Ebeling19, Theorem 1.2]).

For a root lattice L we write $W(L)$ for its Weyl group, which is the Weyl group of the root system $R=L(2)$ (c.f. Appendix B). Of course $W(L)$ is a subgroup of $\operatorname {\mathrm {O}}(L)$ . Similarly we will write $W^a(L)$ for the affine Weyl group of L, which is the semi-direct product

$$\begin{align*}W^a(L) = W(L) \rtimes L \subseteq \operatorname{\mathrm{Iso}}(L) = \operatorname{\mathrm{O}}(L) \rtimes L, \end{align*}$$

where L acts on itself by translations and $\operatorname {\mathrm {Iso}}(L)$ is the affine isometry group of L.

A very nice property of these root systems of type $A_n$ , $D_n$ and $E_n$ is that they are self-dual, that is

$$\begin{align*}A_n = (A_n)^\vee, D_n = (D_n)^\vee, \text{ and } E_n = (E_n)^\vee. \end{align*}$$

So, in particular, if $R = L(2)$ , then $R = R^{\vee }$ (c.f. Appendix B for reference).

From the above discussion we collect information on special points of root lattices, that is points in space which are stabilized by a suitably nice group of affine isometries of the lattice. It turns out that these are precisely the points of the dual of L, which, in general, is the lattice

$$\begin{align*}L^* = \left\{ x \in \mathbb{R}^n : x \cdot v \in \mathbb{Z} \text{ for all } v \in L \right\}. \end{align*}$$

Lemma 3.2. Let L be an n-dimensional root lattice with root system $R = L(2)$ and Weyl group $W(L)$ . For $z \in \mathbb {R}^n$ we have

$$\begin{align*}W^a_z(L) \cong W(L) \quad \Longleftrightarrow \quad z \in L^*. \end{align*}$$

It is the unimodularity of $E_8$ , that is, $E_8 = E_8^*$ , that keeps us from including it in the framework used in Lemma 3.2. In particular we already see that special points for $E_8$ which are not lattice elements will have a stabilizer that is smaller than $W(E_8)$ , we will adress this case in Lemma 3.6.

For the subsequent discussion we need some facts about these lattices, which we collect in Table 1. In particular we emphasise that for indecomposable root lattices not of type $E_8$ all deep holes are elements of the associated dual lattice and therefore special points for the associated affine Weyl group. To justify the contents of this table we refer to Appendix A and more generally to [Reference Conway and Sloane15, Chapter 4].

Table 1 Data regarding indecomposable root lattices.

With this we are ready to discuss the main result of this section.

For all root lattices which are not of type $E_8$ the proof of the above Theorem extends naturally to all special points in Table 1.

3.3 The full stabilizer group for special points and indecomposable root lattices

The proof of Theorem 3.3 relies only on the fact that for root lattices the full stabilizer subgroup of a deep hole contains a Weyl group. We proved the existence of such a subgroup in Lemma 3.2 for all cases except $E_8$ , this case we will handle now. Along with that we compute the full stabilizer subgroups of deep holes in the affine isometry group of indecomposable root lattices. This is achieved in Lemma 3.6. While this is not necessary for the proof of Theorem 3.3 we are not aware of a reference to this problem in the literature, and it seems to be an obvious question how far the full stabilizer of a deep hole in a (indecomposable) root lattice is away from the subgroup identified in Lemma 3.2.

Let $\operatorname {\mathrm {Iso}}(L)$ be the group of affine isometries of L. Every element in $\operatorname {\mathrm {Iso}}(L)$ can be written as the composition of an element $f\in \operatorname {\mathrm {O}}(L)$ by a translation $t_u$ by a vector of $u\in L$ . Recall that for $c\in \mathbb {R}^n$

$$\begin{align*}\operatorname{\mathrm{Iso}}(L)_c = \left\{ \phi \in \operatorname{\mathrm{Iso}}(L) : \phi(c) = c \right\} \subseteq \operatorname{\mathrm{Iso}}(L) \end{align*}$$

be the stabilizer of c in $\operatorname {\mathrm {Iso}}(L)$ .

Firstly, we determine how the translational part and the linear part of an affine isometry $\phi $ are related if $\phi $ is in the stabilizer $\operatorname {\mathrm {Iso}}(L)_c$ of a point c.

Proof. The first assertion is clear since

$$\begin{align*}u = \phi(0) \in L \end{align*}$$

by $\phi \in \operatorname {\mathrm {Iso}}(L)$ . If $c = \phi (c)$ we find

$$\begin{align*}c = \phi(c) = f(c) + u \Rightarrow u = c - f(c). \end{align*}$$

If on the contrary $u = c - f(c)$ then

$$\begin{align*}\phi(c) = f(c) + c - f(c) = c.\\[-34pt] \end{align*}$$

Before determining $\operatorname {\mathrm {Iso}}(L)_c$ , where c is a deep hole of an irreducible root lattice L, we quickly recall the structure of the groups $W(L)$ and $\operatorname {\mathrm {O}}(L)$ for the $A_n$ and $D_n$ series specifically. Note that $D_3 \cong A_3$ , we therefore consider the $D_n$ series only for $n \geq 4$ . In the case of the $A_n$ series we have $W(A_n) \cong S_{n+1}$ , where $S_m$ is the permutation group on m elements, and $\operatorname {\mathrm {O}}(A_n) \cong \left \{ \pm \operatorname {\mathrm {id}} \right \} \times S_{n+1}$ for the full orthogonal group. In the case of the $D_n$ series we have that $W(D_n)$ is generated by all permutations on the n coordinates, together with an even number of sign changes. Abstractly we can write $W(D_n) \cong \left \{ \pm 1 \right \}^n_+ \rtimes S_n \cong \left \{ \pm 1 \right \}^{n-1} \rtimes S_n$ , where $\left \{ \pm 1 \right \}^n_+$ operates by an even number of sign changes on the coordinates. The full orthogonal group is generated by the Weyl group and one additional symmetry and (abstractly) is a semi-direct or wreath product $\operatorname {\mathrm {O}}(D_n) \cong \left \{ \pm 1 \right \}^{n} \rtimes S_{n+1} \cong \left \{ \pm 1 \right \} \wr S_n$ . For the explicit description of $D_n$ as in Appendix A, the additional symmetry separating $\operatorname {\mathrm {O}}(D_n)$ and $W(D_n)$ can be choosen to be the sign change of the last coordinate. For lattices of the $E_n$ -series we have that $\operatorname {\mathrm {O}}(E_7) = W(E_7)$ , while $\operatorname {\mathrm {O}}(E_6) = \left \{ \pm \operatorname {\mathrm {id}} \right \} \times W(E_6)$ .

We also quickly collect some information on the first shell $L(c,\mu (L))$ around a deep hole C in an indecomposable root lattice L, in particular for lattices of type $A_n$ or $D_n$ . Recall that the points $L(c,\mu (L))$ are the vertices of the Delaunay polytope around c.

For $A_n$ a typical deep hole is given by the vector $c = [\lfloor (n+1)/2 \rfloor ]$ (c.f. Appendix A) and the covering radius is $\mu (A_n) = 1/2$ if n is even and $\mu (A_n) = ab/(n+1)$ if n is odd, where $a = \lfloor (n+1)/2\rfloor $ and $b = \lceil (n+1)/2\rceil $ . For $n=2$ we have the hexagonal lattice $A_2$ and the first shell forms the set of vertices of a regular simplex. For n odd the first shell corresponds to the vertices of a centrally symmetric polytope, while this is wrong for n even. To be more precise the vertices of the associated Delaunay polytope are given by the $\binom {n+1}{n+1/2}$ permutations of the vector

$$\begin{align*}( \underbrace{-\tfrac{1}{2},\cdots,-\tfrac{1}{2}}_{\tfrac{n+1}{2} \text{ times}},\underbrace{\tfrac{1}{2},\cdots,\tfrac{1}{2}}_{\tfrac{n+1}{2} \text{ times}}) \in A_n \end{align*}$$

if n is odd, and the $\binom {n+1}{a}$ permutations of the vector

$$\begin{align*}( \underbrace{-\tfrac{b}{n+1},\cdots,-\tfrac{b}{n+1}}_{a \text{ times}},\underbrace{\tfrac{a}{n+1},\cdots,\tfrac{a}{n+1}}_{b \text{ times}}) \in A_n \end{align*}$$

if n is even. We refer to [Reference Conway and Sloane16, Section 4] for more details.

For $D_n$ , we can describe the Delaunay polytopes around all holes in terms of the representatives $[1],[2],[3]$ as given in Appendix A. The Delaunay polytopes around $[1]$ and $[3]$ are half-cubes (also known as demi-cubes or parity-polytopes) which are centrally symmetric if and only if n is even. For $[2]$ the Delaunay polytope is a cross-polytope, which is centrally symmetric for arbitary n. Now for $n=3$ there is one type of deep hole, represented by $[2]$ , for which the Delaunay polytope is a cross-polytope. For $n=4$ there are three types of deep holes, represented by $[1],[2],[3]$ and in this case the half-cube and cross-polytope coincide. For $n \geq 5$ there are two types of deep holes, represented by $[1]$ and $[3]$ and in these cases the Delaunay polytopes are half-cubes.

For the $E_8$ lattice the Delaunay polytope around a deep hole is a cross polytope. The Delaunay polytope at a deep hole of $E_6$ is a polytope with $27$ vertices, known as Schläfli polytope $2_{21}$ . The Delaunay polytope at a deep hole of $E_7$ is a centrally symmetric polytope with $56$ vertices, known as Hesse polytope $3_{21}$ . A justification of the above and more details on the geometric descriptions of these polytopes can be found in [Reference Conway and Sloane16].

Proof. We prove the statements by using explicit data on the indecomposable root lattices and their Weyl and automorphism groups.

We start with $L \cong E_8$ Footnote ⁴ and use the explicit realisation of $E_8$ as in Appendix A, given by

(6) $$ \begin{align} E_8 = D_8^+ = D_8 \cup \tfrac{1}{2}e + D_8. \end{align} $$

Let c be a deep hole of $E_8$ for which $0$ is a closest lattice point. Then $\|c\|^2 = 1$ and the associated Delaunay polytope D is a cross polytope.

If $\phi \in \operatorname {\mathrm {Iso}}(L)_c$ then $\phi $ permutes the vertices of D and therefore fixes D. We identify $\operatorname {\mathrm {Iso}}(L)_c$ with a subgroup $G'$ of $\operatorname {\mathrm {O}}(\mathbb {R}^8)$ . The inner automorphism $\Phi : \operatorname {\mathrm {Iso}}(\mathbb {R}^8) \rightarrow \operatorname {\mathrm {Iso}}(\mathbb {R}^8); \phi \mapsto \varphi = t_{-c} \circ \phi \circ t_{c}$ induces the isomorphism

$$\begin{align*}\operatorname{\mathrm{Iso}}(L)_c \cong G' = t_{-c} \circ \operatorname{\mathrm{Iso}}(L)_c \circ t_{c}, \end{align*}$$

where $G'$ is a subgroup of $\operatorname {\mathrm {O}}(\mathbb {R}^8)$ as desired. In agreement with Lemma 3.5 we explicity obtain

(7) $$ \begin{align} G' = \left\{ \varphi \in \operatorname{\mathrm{O}}(E_8) : \varphi(c)-c \in E_8 \right\}. \end{align} $$

We claim that $G' \cong W(D_8)$ .

For this we fix the deep hole $c = e_1$ , for which the Delaunay polytope is

$$\begin{align*}D = e_1 + \operatorname{\mathrm{conv}}\left(\left\{ \pm e_1, \ldots, \pm e_8 \right\} \right) = e_1 + D'. \end{align*}$$

Let $\phi \in \operatorname {\mathrm {Iso}}(L)_{e_1}$ , so it stabilizes D and its centre $e_1$ , and so $\varphi = t_{-e_1} \circ \phi \circ t_{e_1}$ stabilizes $D' = -e_1 + D$ . This is equivalent to

$$\begin{align*}\varphi\left(\left\{ \pm e_1,\ldots, \pm e_8 \right\}\right) = \left\{ \pm e_1,\ldots, \pm e_8 \right\} \end{align*}$$

so $\varphi $ stabilizes a standard cross polytope centered at the origin. The group of isometries of the latter is precisely the Weyl group of $D_8$ , which one can see for example by the description $W(D_8) \cong \left \{ \pm 1 \right \}^8_+ \rtimes S_8$ .

On the contrary assume that $\varphi \in W(D_8)$ . Then, in (6), $\varphi $ fixes the sublattice $D_8$ of $E_8$ and operates on e by flipping an even number of signs. This gives $\varphi (e)-e \in \left \{ 0,2 \right \}^8$ , where an even number of $2$ s occur, so $\varphi (e)-e \in 2D_8$ . With this and the decomposition (6) we see that $\varphi $ maps $E_8$ to $E_8$ , so $\varphi \in \operatorname {\mathrm {O}}(E_8)$ . Moreover

$$\begin{align*}\varphi(e_1) = \pm e_i \quad \Rightarrow \quad \varphi(e_1) - e_1 = \pm e_i -e_1 \in E_8, \end{align*}$$

so $\varphi \in G'$ as defined by (7). This finishes the proof for $L \cong E_8$ .

We recall that if c is a deep hole of an irreducible root lattice L of type other than $E_8$ , then $c \in L^*$ . Furthermore, we recall that in these cases there is an isomorphic copy of $W(L)$ contained in the stabilizer of c in $W^a(L) \subseteq \operatorname {\mathrm {Iso}}(L)$ , that is, $W(L) \hookrightarrow \operatorname {\mathrm {Iso}}(L)_c$ . Indeed this immediately follows from Lemma 3.2, which asserts that the stabilizer of c in $W^a$ is isomorphic to $W(L)$ as in the proof of Theorem 3.3.

For $L \cong A_n$ we again use the inner automorphism $\Phi : \operatorname {\mathrm {Iso}}(\mathbb {R}^n) \rightarrow \operatorname {\mathrm {Iso}}(\mathbb {R}^n); \phi \mapsto \varphi = t_{-c} \circ \phi \circ t_{c}$ to obtain an isomorphism

$$\begin{align*}\operatorname{\mathrm{Iso}}(L)_c \cong G' = t_{-c} \circ \operatorname{\mathrm{Iso}}(L)_c \circ t_{c}, \end{align*}$$

so we can work with a group $G'$ of linear isometries, with $W(L) \subseteq G'$ . Let $D = c + D'$ be the Delaunay polytope of c. So if $\phi $ stabilizes D, $\Phi (\phi )$ stabilizes $D'$ and vice versa.

We recall that $\operatorname {\mathrm {O}}(A_n) = \left \{ \pm \operatorname {\mathrm {id}} \right \} \times W(A_n)$ . If n is even, then $D'$ is not centrally symmetric and so $G'$ cannot contain $-\operatorname {\mathrm {id}}$ , therefore $\operatorname {\mathrm {Iso}}(L)_c \cong G' = W(A_n)$ . If n is odd, then $D'$ is centrally symmetric and one can check that indeed $-\operatorname {\mathrm {id}} \in G'$ , so $\operatorname {\mathrm {Iso}}(L)_c \cong G' = \operatorname {\mathrm {O}}(A_n)$ .

For $L \cong D_n$ we distinguish between $n = 4$ and $n \geq 5$ . The case $n=4$ can be handled by direct computations using the data in Appendix A or using a computer algebra program such as MAGMA, we do not include the details since the latter is easily doneFootnote ⁵ .

The case $n\geq 5$ : Here $D_n$ has two types of deep holes which are inequivalent under translations by lattice elements of $D_n$ . Explicitly we can write

$$\begin{align*}D_n^* = \left([0] + D_n\right) \cup \left([1] + D_n\right) \cup \left([2] + D_n\right) \cup \left([3] + D_n\right), \end{align*}$$

where $[1],[2],[3]$ are holes of $D_n$ given by

$$ \begin{align*} [0] &= (0,0,\ldots,0) \text{ of squared norm } 0\\ [1] &= (\tfrac{1}{2},\tfrac{1}{2},\ldots,\tfrac{1}{2}) \text{ of squared norm } n/4\\ [2] &= (0,0,\ldots,1) \text{ of squared norm } 1\\ [3] &= (\tfrac{1}{2},\tfrac{1}{2},\ldots,-\tfrac{1}{2}) \text{ of squared norm } n/4. \end{align*} $$

Note that for $n \geq 5$ the two types of deep holes are $[1]$ and $[3]$ . Now $\operatorname {\mathrm {O}}(D_n)$ is generated by $W(D_n)$ and an isometry interchanging $[1]$ and $[3]$ , but this additional isometry cannot be contained in $G'$ and so $G' = W(D_n)$ . So $\operatorname {\mathrm {Iso}}(L)_c \cong G' = W(D_n)$ .

For $E_7$ we note that $\operatorname {\mathrm {O}}(L) = W(L)$ and $E_6$ can be handled by explicit computations or by using a computer algebra program, such as Magma, we omit the details.

It might be of interest that a partial reverse holds. If z is a point, for which the stabilizer $\operatorname {\mathrm {Iso}}(L)_z$ of z in $\operatorname {\mathrm {Iso}}(L)$ contains the Weyl group of the root system of L, then z is either a hole of L (if $L \not \cong E_8$ ) or an element of L.

Corollary 3.7. For an irreducible root lattice $L \not \cong E_8$ , we have

$$\begin{align*}W(L) \hookrightarrow \operatorname{\mathrm{Iso}}(L)_z \quad \Longleftrightarrow \quad z \text{ is a hole or an element of } L. \end{align*}$$

Proof. The argument is as in the proof of Lemma 3.6, we get the embedding of $W(L)$ into the stabilizer by Lemma 3.2 for all points of the dual lattice $L^*$ . We then use the coset-decomposition of $L^*$ as in Appendix A, from which we infer that every element of $L^*$ is either a lattice point or a hole of L.

Explicitly for $A_n$ we have the decomposition

$$\begin{align*}A_n^* = \left([0] + A_n\right) \cup \left([1] + A_n\right) \cup \cdots \cup \left([n] + A_n\right), \end{align*}$$

where $[0],[1],\ldots ,[n]$ are the vertices of a fundamental simplex for $W^a(A_n)$ . The elements $[1],\ldots ,[n]$ are precisely the holes of $A_n$ (up to translations by lattice elements).

For $D_n$ we have the decomposition

$$\begin{align*}D_n^* = \left([0] + D_n\right) \cup \left([1] + D_n\right) \cup \left([2] + D_n\right) \cup \left([3] + D_n\right), \end{align*}$$

where $[1],[2],[3]$ are the holes of $D_n$ .

For $E_6$ and $E_7$ it follows similarly, compare the data in Appendix A.

3.4 Finding explicit $\alpha $

In order to find an explicit $\alpha _0$ , we need to make explicit the covering strategy of the proof of Theorem 1.2 described in Section 2.3.

In the case of root lattices, the Voronoi cell is the union of the images of the fundamental simplex under the action of the Weyl group. In the two examples we consider next, we cover this fundamental simplex with exactly two balls. Here is the strategy we follow in order to show that a given $\alpha _0$ works. First, $\alpha _0$ has to fulfill the condition of Lemma 2.9, and this lemma gives a radius $R_{\alpha _0}$ such that for every $\alpha \geq \alpha _0$ , the deep hole c is the unique minimizer for the potential in the ball $B(c,R_{\alpha _0})$ . Then, the radius $\varrho $ of the second ball $B(0,\varrho )$ is the largest norm among the points in the fundamental simplex that are not covered by $B(c,R_{\alpha _0})$ . Denote by S the intersection of the edges of the simplex with the sphere of radius $R_{\alpha _0}$ centered at c, and decompose the set of vertices of the fundamental simplex as $V\cup V'$ , where the vertices in V belong to $B(c,R_{\alpha _0})$ , and those in $V'$ do not. Then $B(c,R_{\alpha _0})$ covers the convex hull of $S\cup V$ , so it is enough that $B(0,\varrho )$ covers the convex hull of $S\cup V'$ , see Figure 5 for two examples in dimension $2$ . It remains to check that $\alpha _0 \geq \alpha _\varrho $ , where $\alpha _\varrho $ is given by Theorem 2.1, which, according to Lemma 2.2, amounts to verifying the inequality

$$\begin{align*}e^{-\alpha_0(\mu(L)^2 - \varrho^2)} \left(\frac{2\alpha_0 \mu(L)^2 e}{n}\right)^{n/2} < 1. \end{align*}$$

Figure 5 Covering a simplex with two spheres in dimension 2.

Using this strategy, we can prove:

Proof. First note that $\tau _L = 1$ , and that all the shells around c are centrally symmetric. One can then check that $\alpha _0 = 23$ satisfies the conditions of Lemma 2.9, which gives $R_{\alpha _0}>0.66$ . Then, the vertices of the fundamental simplex are $0$ and

$$\begin{align*}\begin{array}{lcl} v_1 =(-\frac{1}{8},\frac{1}{8},\frac{1}{8},\frac{1}{8},\frac{1}{8},\frac{1}{8},\frac{1}{8},\frac{7}{8}) & \quad & v_2 = (0,0,\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{5}{6}) \\ v_3 = (\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{5}{6}) & \quad & v_4 = (0,0,0,\frac{1}{5},\frac{1}{5},\frac{1}{5},\frac{1}{5},\frac{4}{5}) \\ v_5 = (0,0,0,0,\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{3}{4}) & \quad & v_6 = (0,0,0,0,0,\frac{1}{3},\frac{1}{3},\frac{2}{3}) \\ v_7 = (0,0,0,0,0,0,\frac{1}{2},\frac{1}{2}) & \quad & v_8 = (0,0,0,0,0,0,0,1) \end{array} \end{align*}$$

where $v_8=c$ is a deep hole of L. Except $v_7$ and $0$ , all the vertices are covered by $B(c,0.66)$ . Therefore, the sphere centered at c and of radius $0.66$ intersects the edges $[0,v_i]$ with $i\neq 7$ , and the edges $[v_7,v_i]$ , $i \neq 7$ . Computing these intersections, we find that the point with the largest norm lies on the edge $[v_6,v_7]$ , and has norm smaller than $\varrho = 0.72$ . Since we also have $\| v_7 \| < \varrho $ , the fundamental simplex is covered by the union of $B(c,0.66)$ and $B(0,\varrho )$ . To conclude, we check that for $\alpha _0 = 23$ ,

$$\begin{align*}e^{- \alpha_0 (1 - \varrho^2)} \left(\frac{ e \alpha_0}{4}\right)^4 < 1.\\[-34pt] \end{align*}$$

In a similar way, we get