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 results, 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.

Start with $K = \mathbb{Q}[\mu_n]$ , is the $n$th cyclotomic number field. Then $K_{\R}= K\otimes_{\mathbb{Q}}\R$ is a $\varphi(n)$-dimensional real vector space. Set $d = 2 \varphi(n)$

Take $\R^{d} \simeq K_{\R}^{\oplus 2}$. Identify $\Z^{2\varphi(n)}$ with $\mathcal{O}_K^{\oplus 2}$, two copies of the ring of integers in this vector space. Take $G = \Z/n\Z$ acting on this lattice via $\langle \mu_n \rangle$, multiplication by the $n$th roots of unity.

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 $n$!

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 $\Gamma$.

André Weil discussed a much more general formula in 1965.

His formula is an an "adelic" setting, but it can be used to derive integration all the integration formulas that have been used so far!

In particular, one can also derive the higher-moment formulas due to Rogers.

\(\DeclareMathOperator{\vol}{vol}\)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 $n \ll \sqrt{d}$, the higher moments start behaving like moments of a Poisson distribution.

Incidentally, the $\mathcal{O}_{K}$-lattices considered by Venkatesh are also used in coding theory and cryptography.

We fix $t \in \mathbb{Z}_{\ge 2}$ and look at the following random set of lattices. $$ \{ g \cdot \mathcal{O}_K^{\oplus t} \ | \ g \in SL_t(K_{\R}) \} \simeq SL_t(K_\mathbb{R} ) /SL_t(\mathcal{O}_K). $$
We call these lattices random $\mathcal{O}_K$ lattices.

Each $\mathcal{O}_K$ lattice $\Lambda \subseteq K_\mathbb{R}^t$ lives in a real space of dimension $t \times [K:\mathbb{Q}]$. We equip $K_\mathbb{R}^t$ with the "normalized trace form" so that each $\Lambda$ has unit covolume. On $K_{\mathbb{R}}$ it is \begin{equation} \langle x,y \rangle = \Delta_K^{-\frac{2}{[K:\mathbb{Q}]}} \text{Tr}(x\overline{y}). \end{equation} On $K_{\mathbb{R}}^t$, it is extended by just summing up "coordinate-wise".

Then does Poisson like behaviour also exist for this specific class of random lattices? We are interested in $t \rightarrow \infty$ and, also the more difficult scenario, $[K:\mathbb{Q}] \rightarrow \infty$.

A class of number fields $\{ K_i \}_{i \in I}$ satisfies the Bogomolov property if there exists $c>0$ such that for each $\alpha \in \bigcup_i K_i$, either $\alpha$ is a root of unity or $$h(\alpha) = \tfrac{1}{\deg(\alpha)} \sum_{v \in \cal{M}_{\mathbb{Q}(\alpha) } } \max\{0, \log|\alpha|_v\} \geq c.$$

The Bogomolov property is satisfied by the class of cyclotomic number fields, the class of abelian number fields, and most importantly, for a fixed number field (so $t$ can grow and $K$ can be fixed).

But there are classes of number fields where the Bogomolov property fails. In fact, we can show that for one such class of number fields, the second moment is not the Poisson distribution.

First you need a higher moment formula. For this, we can use Weil's work (difficult to read) or some recent preprints by S. Kim or N. Hughes or my doctoral thesis.

What follows is many pages of calculations!

We split the matrices into four different types of matrices. One type of matrices are the main terms, the rest are error terms.

\(\DeclareMathOperator{\OK}{\mathcal{O}_K}\)Here are some of the calculations for the second moment. Let $f = \mathbb{I}_B$ be the indicator function of an origin-centered ball $B$. The integration formula first gives that our expected value is $$\vol(B)^2+\sum_{\alpha\in K^\times}[\OK:\alpha^{-1}\OK \cap\OK]^{-t}\cdot\vol(B\cap\alpha B).$$

To arrive at our result it suffices to prove exponential decay of the sum \begin{equation} \sum_{\alpha\in (K^\times\setminus\mu_K)/\mu_K}[\OK:\alpha^{-1} \OK \cap\OK]^{-t}\cdot\frac{\vol(B\cap\alpha B)}{\vol(B)}, \end{equation}

The main terms are then contributed by $\alpha \in \langle \mu_n \rangle$.

For the error terms, we write $S_\beta=\sum_{\alpha\in (\OK^\times\setminus\mu_K)/\mu_K}\frac{\vol(B\cap\alpha\beta B)}{\vol(B)}$ and evaluate $$\sum_{\beta \in (K^{\times}\setminus \OK^{\times})/\OK^{\times}} [\OK:\beta^{-1} \OK \cap\OK]^{-t} S_{\beta}.$$

We then approximate $$\frac{\vol(B\cap\alpha\beta B)}{\vol(B)} \leq \frac{1}{\cosh( 2 \cdot h(\alpha \beta))^{t[K:\mathbb{Q}]/2 }}$$ using the convex combinations idea. Here the Bogomolov property helps to prove exponential decay.

And then it remains to put a strict inequality to bound the point counts $$\# \{ \alpha \in \OK^{\times} \mid h(\alpha \beta ) \leq T\}$$ in terms of $\beta,T$. This is done using a packing argument in the log embedding of units.

Then, combine the point counts with exponential decay to get $$ S_{\beta} \ll N(\beta)^{-t \varepsilon_1} \cdot \exp( \varepsilon_2 (t[K:\mathbb{Q}])$$

Keeping track of the constants, we can bound the sum of $S_{\beta}$ over $\beta$ with a special value of Dedekind zeta function.