The following theorem is called Siegel’s mean value theorem.
(Siegel, 1945, [1]) Suppose \(f:\mathbb{R}^{d} \rightarrow \mathbb{R}\) is a compactly supported continuous function. Then, the following holds. \[\begin{align} \int_{SL_d(\mathbb{R})/SL_d(\mathbb{Z})}^{} \left( \sum_{v \in g \mathbb{Z}^{n} \setminus \{ 0\}}^{}f(v) \right) dg = \int_{\mathbb{R}^{n}}^{} f(x) dx, \end{align}\] where the \(dx\) on the RHS is the usual Lebesgue measure on \(\mathbb{R}^{d}\).
Setting \(f\) as the indicator function of a ball, the left-hand side gives us the average intersection number of a ball and a lattice. This gives the following lower bound on the sphere packing constant. \[\begin{align} c_d \ge 2. \end{align}\]
In the same paper as the mean value theorem, Siegel also showed the following.
(Siegel, 1945, [1]) Suppose \(f:\mathbb{R}^{d} \rightarrow \mathbb{R}\) is a compactly supported continuous function. Then, the following holds. \[\begin{align} \int_{SL_d(\mathbb{R})/SL_d(\mathbb{Z})}^{} \left( \sum_{\substack{ v \in g \mathbb{Z}^{n} \setminus \{ 0\} \\ v = (v_1,v_2,\ldots,v_n) \\ \gcd(v_1,v_2,\ldots,v_n)=1 } }^{}f(v) \right) dg = \frac{1}{\zeta(k)} \int_{\mathbb{R}^{n}}^{} f(x) dx, \end{align}\] where the \(dx\) on the RHS is the usual Lebesgue measure on \(\mathbb{R}^{d}\).
This gives the following bound on the sphere packing constant \[\begin{align} c_d \ge 2 \zeta(d). \end{align}\]
Here \(\zeta\) is the Riemann-zeta function. Note that \(\zeta(d) \rightarrow 1\) as \(d \to \infty\).
(Venkatesh, 2013, [2]) Let \(d = 2\varphi(n)\) and \(K = \mathbb{Q}(\mu_{n})\). Suppose \(f:K_{\mathbb{R}}^2 \rightarrow \mathbb{R}\) is a compactly supported bounded measurable function. Then, the following holds. \[\begin{align} \int_{SL_2(K_\mathbb{R})/SL_2(\mathcal{O}_K)}^{} \left( \sum_{v \in g \mathcal{O}_K^{\oplus 2} \setminus \{ 0\}}^{}f(v) \right) dg = \int_{\mathbb{R}^{d}}^{} f(x) dx, \end{align}\] where the \(dx\) on the right-hand side is that Lebesgue measure on \(\mathbb{R}^{d}\) that makes \(\mathcal{O}_K^{\oplus 2} \subseteq K_\mathbb{R}^2 \simeq \mathbb{R}^d\) of unit covolume and \(dg\) is the unique \(SL_2(K_\mathbb{R})\)-invariant probability measure on \(SL_2(K_{\mathbb{R}})/SL_2(\mathcal{O}_{d})\).
Setting \(f\) as the indicator function of a ball, the left-hand side gives us the average intersection number of a ball and a lattice. This gives the following lower bound on the sphere packing constant. \[\begin{align} c_{2 \varphi(n)} \ge n. \end{align}\]
Does there exist a primitive vectors version of Venkatesh’s mean value theorem? Can it be used to prove something like \(c_{2 \varphi(n)} \ge n \zeta_{K}(2)\) for \(K = \mathbb{Q}(\mu_{n})\)?
Here \(\zeta_K\) is the Dedekind-zeta function
This was shown in [3]
Let \(D\) be a \(\mathbb{Q}\)-division algebra containing an order \(\mathcal{O} \subseteq D\). Let \(G = SL_k(D_{\mathbb{R}})\) and \(\Gamma = SL_k(\mathcal{O})\), for some \(k \ge 2\). Let \(dg\) be the probability measure on \(G/\Gamma\) that is left-invariant under \(G\) action. Then for any \(f \in C_{c} (D_{\mathbb{R}}^{k})\), we obtain that \[\begin{align} \int_{G / \Gamma}\left( \sum_{v \in g \mathcal{O}^{k} \setminus \{ 0\}} f(v) \right) dg = \int_{D_{\mathbb{R}}^{k}}^{} f(x) dx, \end{align}\] where \(dx\) is a Lebesgue measure on \(D_{\mathbb{R}}^{k}\) with respect to which \(\mathcal{O}^{k}\) has a covolume of \(1\).
This page was updated on August 24, 2022.
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