Random Arithmetic Lattices as Sphere Packings

The exact value of the constant $c_d$ is known only for $d = \{ 1,2,3,4,5,6,7,8,24\}$. For other $d$, we want to understand the asymptotic behaviour. In this talk, we will only focus on lower bounds.

Since $\lim \inf \left( \frac{\varphi(n)}{n} \log \log n\right) = e^{- \gamma}$, the last bound is the best lower bound (among these, and overall) on $c_d$ in infinitely many dimensions. The first dimension where it outperforms all others in this list is $d=960$.

To recover Venkatesh's result, set $D=\mathbb{Q}(\mu_n)$, $\mathcal{O} = \mathbb{Z}[\mu_n]$ and $G_0 = \langle \mu_n \rangle$. Hence, this gives \begin{align} c_{2 \varphi(n)} \ge n \end{align}

The cherrypicked sequence of Venkatesh achieves an asymptotic growth of $O( d \log \log d )$. This is achieved by setting $K$ as the $n$th cyclotomic field where $n=\prod_{p < N} p$.

The division algbera construction gives more freedom to cherrypick sequences. Instead of choosing a sequence of cyclotomic fields, we can now choose sequences of $\mathbb{Q}$-divison algebras. However, no such sequence will be able to give an asymptotic result strictly better than $O( d \log \log d )$. Improvements in individual dimensions is still possible, as shown before.

To show $c_d \ge K$, we must prove the existence of $g \in SL_d(\R)$ such that the origin centered ball $B$ with $\mu(B) = K$ has $g \Z^d \cap B = \{0\} $

This is an optimization problem on the space \[ X_d:= \{ \Lambda \subset \mathbb{R}^{d} | \ \mu(\mathbb{R}^{d} / \Lambda) = 1 \} = \{ g\Z^d \ | \ g \in SL_d(\R) \} \]

\[ \simeq SL_d(\mathbb{R})/SL_d(\mathbb{Z}). \]

A priori, this is a bijection of sets. But now we can pull back the topology and the measure from $SL_d(\R)/SL_d(\Z)$.

$SL_d(\R)$ has the topology of a locally compact group.

$SL_d(\R)$ is unimodular. $SL_d(\Z)$ is a discrete subgroup inside $SL_d(\R)$ and therefore there is a unique left $SL_d(\R)$-invariant measure on $SL_d(\R)/SL_d(\Z)$.

This gives us a probability space. Hence we can talk about random unit covolume lattices.

What we are empirically confirming is the following.

But as you saw that for any $\Lambda \in X_d$, when $f$ is the indicator of a ball, we must have $\Phi_f(\Lambda) \in \{0,2,4,6,...\}$. That's because balls are symmetric. $$v \in \text{supp}(f) \cap ( \Lambda \setminus \{0 \}) \Rightarrow -v \in \text{supp}(f) \cap ( \Lambda \setminus \{0 \}).$$

If $f$ is the indicator function of a ball of volume $2- \varepsilon$, then this tells us that for any dimension $d$ $$\int_{X_d} \Phi_f = 2 - \varepsilon$$

Conclusion: There exists some lattice $\Lambda \in X_d$ such that $\Phi_f(\Lambda) = 0$. That is, there is some lattice of unit covolume that intersects trivially with an origin centered ball of volume $ 2 - \varepsilon$.

Another conclusion: $c_d \ge 2$ for all dimensions $d$!

Both Venkatesh and the division algebra lattices use this idea. What we want is to find expectation of the lattice sum over a smaller subset of lattices that have a larger group of symmetries.

For Venkatesh, the group of symmetries is always a cyclic group. For the new result, the symmetries are non-commutative.

The idea is to take a nice enough subcollection $Y_d \subseteq X_d$ of lattices and average the lattice-sum function $\Phi_f$ over them.

Define the set $Y_d$ as $$Y_d = \left\{ \left[ \begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right] \mathcal{O}_K^2 \ | \ a,b,c,d \in K_{\R} = K\otimes \R , ad-bc =1_{K_{\R}} \right\}.$$

In conventional notation, this is just $$Y_d = \{ g ( \mathcal{O}_K^{\oplus 2} ) \ | \ g \in SL_2(K_{\R}) \} \simeq SL_2(K_\mathbb{R} ) /SL_2(\mathcal{O}_K). $$

$Y_d$ can be given probability measure.

So if $f$ is the indicator function of an origin-centered ball with respect to a quadratic form that is invariant under $\langle \mu_n \rangle$, we have that $\Phi_f(\Lambda) \in \{0,n,2n,3n,4n,\ldots\}$ for $\Lambda \in Y_d$. Such a quadratic form always exists by averaging!

Venkatesh, then proves the following analogue of Siegel's theorem.

Conclusion: By setting $f$ as the indicator function of a ball in a suitable quadratic form, we conclude that there exists some lattice $\Lambda \in Y_d$ such that $\Phi_f(\Lambda) = 0$. That is, there is some lattice of unit covolume that intersects trivially with an origin centered ball of volume $ n - \varepsilon$.

Another conclusion: $c_{2 \varphi(n)} \ge n$ for all $n$!

The division algebra case is also very similar. Let $D$ be a finite-dimensional division algebra over $\mathbb{Q}$. Let $\mathcal{O} \subseteq D$ be an order in $D$. We work in $d = 2 \dim_{\mathbb{Q}} D$ dimensions. Define $D_{\R}= D \otimes_\mathbb{Q} \R$. $$Y_d = \{ g ( \mathcal{O}^{\oplus 2} ) \ | \ g \in SL_2(D_{\R}) \} \simeq SL_2(D_\mathbb{R} ) /SL_2(\mathcal{O}). $$
Here $$ SL_2(D_{\R}) = \left\{ \left[ \begin{matrix} a & b \\ c & d \end{matrix}\right] \ | \ \left[ \begin{matrix} x \\ y \end{matrix}\right] \mapsto \left[ \begin{matrix} ax + by \\ cx + dy \end{matrix}\right] \text{ is a measure preserving map on } D_{\R}^{\oplus 2} \right\}$$

$Y_d$ consists of lattices that are invariant under diagonal right-multiplication by units in $\mathcal{O}$ $$ g ( \mathcal{O}^{\oplus 2} ) = g ( \mathcal{O}^{\oplus 2} ) \left[ \begin{matrix} \mu & 0 \\ 0 & \mu \end{matrix}\right] , \text{ for any } \mu \in \mathcal{O}^{*} $$

To get packing bounds, fix a finite subgroup $G_0 \subseteq \mathcal{O}^*$ to act diagonally on the right of $\mathcal{O}^{\oplus 2}$. We get the bounds $$c_{2 \dim_{\mathbb{Q}} D} \ge \# G_0.$$

In fact we only need to find finite subgroups that live in $\mathbb{Q}$-division algebras. The order $\mathcal{O}$ can be aligned according to the finite group. Fortunately, there exists a complete classification of such finite subgroups due to Amitsur, 1955.

For brewity, we will not discuss classification of of finite subgroups of division rings.

What about other homogeneous spaces of lattices? All of the above integral formulas were on spaces $\mathcal{G}(\mathbb{R})/ \Gamma$ for some algebraic $\mathbb{Q}$-group $\mathcal{G}$ and an arithmetic subgroup $\Gamma$.

If the group $\mathcal{G}$ is semisimple, we are automatically guaranteed that $\mathcal{G}(\mathbb{R})/ \Gamma$ will be a probability space under the natural left $\mathcal{G}(\mathbb{R})$-invariant Haar measure

André Weil discussed a much more general formula in 1965.

His formula is an an "adelic" setting, but let us try to interpret it for our use.

• Let $\pi :\mathcal{G} \rightarrow GL(V)$ be a $\mathbb{Q}$-representation of a semisimple $\mathbb{Q}$-algebraic group.
• Let $\Lambda \subseteq V_\mathbb{Q}$ be a maximal-rank base lattice on which the group will act to randomize it.
• Let $\Gamma \subseteq \mathcal{G}(\mathbb{Q}) = \{ g \in \mathcal{G}(\mathbb{Q}) \ | \ g \Lambda = \Lambda\}$
• Let $f:V_\mathbb{R} \to \mathbb{R}$ be a test function.

Then assuming that the left side below converges absolutely, Weil's formula is \[ \DeclareMathOperator{\vol}{vol} \int_{\mathcal{G}(\mathbb{R})/\Gamma} \left( \sum_{v \in g\Lambda} f(v) \right) dg = \sum_{ \Gamma w \in \Gamma \backslash \Lambda } \vol\left( \mathcal{G}_w(\mathbb{R})/\Gamma_w \right) \int_{ g \in \mathcal{G}(\mathbb{R}) / \mathcal{G}_{w}(\mathbb{R})} f(gx)dg, \] where \begin{align} \mathcal{G}_w(\mathbb{Q}) & = \{ g \in \mathcal{G}(\mathbb{Q}) \ | \ gw = w\}, \\ \Gamma & = \{ g \in \Gamma \ | \ gw = w\}, \end{align}

Weil also gives some sufficient conditions for when the left-hand side will exist. However, at this level of abstraction this is a bit difficult to handle.

But observe that in the innermost term, we are just integrating $f$ along an orbit $\mathcal{G}(\mathbb{R}) w$ and then summing up over lattice orbits. This is all we need to start computing!

Let us work with one very concrete example in this framework. Suppose we want to computer higher moments of our Siegel transforms in the most classical case. That is, we want to evaluate for some $n \ge 2$ \begin{equation} \int_{SL_d(\mathbb{R})/SL_d(\mathbb{Z})} \left( \sum_{v \in g \mathbb{Z}^{d} } f(v) \right)^{n} dg = \int_{SL_d(\mathbb{R})/SL_d(\mathbb{Z})} \left( \sum_{ (v_1,v_2,\dots,v_n) \in (g \mathbb{Z}^{d})^{n} } f(v_1) f(v_2) \dots f(v_n) \right) dg \end{equation}

For this, we can use Weil's setup as follows:

• $ \pi : SL_d \rightarrow GL_{d \times n }$ is the diagonal action on $(\mathbb{Q}^{d})^{\oplus n}$,
  
• $(\mathbb{Z}^{d} )^{n } \subseteq \mathbb{R}^{d \times n } $ is the base lattice,
  
• $SL_d(\mathbb{Z})$ stabilizes $\mathbb{Z}^{d \times n }$ under the left action (or equiv. diagonal action on $(\mathbb{Z}^d)^{\oplus n}$),
• Test function is $g : \mathbb{R}^{d \times n } \rightarrow \mathbb{R}$ as $g = f^{\otimes n }$.

Then, we get \begin{equation} \int_{SL_d(\mathbb{R})/SL_d(\mathbb{Z})} \left( \sum_{v \in g \mathbb{Z}^{d} } f(v) \right)^{n} dg = \sum_{ \text{orbit}~w\in SL_d(\mathbb{Z}) \backslash \mathbb{Z}^{d \times n }} \cdots \end{equation}

But then what are the orbits of $SL_d(\mathbb{Z})$ action on $\mathbb{Z}^{d \times n}$? Let us work this out for $n < d$.

Let $\alpha_1, \alpha_2 ,\dots, \alpha_n \in \mathbb{Q}^{n}$. Then for $v_1, v_2 ,\dots,v_n \in \mathbb{Z}^{d}$ and $g \in SL_d(\mathbb{Z})$, we have \begin{align} \alpha_1 v_1 + \alpha_2 v_2 + \dots + \alpha_n v_n = 0 \Rightarrow \alpha_1 g v_1 + \alpha_2 g v_2 + \dots \alpha_n g v_n = 0. \end{align}

If $v_1,\dots,v_n$ span an $m$-dimensional space, they will have $n-m$ linear dependencies. These linear dependence coeffiencients $(\alpha_1, \alpha_2,\dots, \alpha_n) $ span an $(n-m)$-dimensional space in $\mathbb{Q}^{n}$.

This almost determines all the orbits. For $n < d$, the orbits of $SL_d(\mathbb{Z})$ action on $(\mathbb{Z}^{d} \setminus \{0\})^{n}$ are in bijection with subpaces of $\mathbb{Q}^{n}$.

We are now ready to state a higher moment formula due to Rogers (1966).

For any $\mathbb{Q}$-subspace $ S \subseteq \mathbb{Q}^{n}$, we can define a height $H(S) $ as \begin{equation} H(S) = \vol\left( \frac{S \otimes_\mathbb{Q} \mathbb{R}}{ S \cap \mathbb{Z}^{n}} \right), \end{equation} the volume taken with respect to natural measure for subpaces of $\mathbb{R}^{n}$.

Why is it a finite number?

The formula basically writes the higher moment as a "height zeta function" value. In fact the fact that the right hand side exists is the equivalent to saying that the height zeta functions on Grassmannian varieties absolutely converge at special values.

Knowing higher moments of Siegel gives more information about random lattices!

When $n \ll d$, the higher moments start behaving like moments of a Poisson distribution.

Rogers used this to prove that \begin{equation} c_d \ge \tfrac{1}{3}\sqrt{d}. \end{equation}

But he had already obtained in 1947 \begin{equation} c_d \ge d \cdot ( 2e^{-1} ), \end{equation} using just the first moment!

This is because the first moment gives a much simpler expression. Indeed, $SL_d(\mathbb{Z}) \in \mathbb{Z}^{d}$ has only two orbits (and not infinitely many!).

Hopefully, we will discover better ways to handle these higher moment formulae without having to approximate them.

Or perhaps we will be able to find better algebraic groups for which the integration formulas are more tractable.

The quest to improve lower bounds on sphere packing goes on!