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Let $\mu$ be the Haar measure defined on the space of unimodular lattices, identified with $\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$.

The classical Siegel's formula in geometry of numbers states that for $f\in L^1(\mathbb R^d)$, let $\Lambda$ be a unimodular lattice, and let

$$\hat{f}(\Lambda):= \sum_{v\in \Lambda\setminus 0} f(v).$$

Then $$\int_X \hat{f} d\mu = \int_{\mathbb R^d} f dv.$$

I wonder if there is such a generalization to this result:

For $g\in L^1((\mathbb R^d)^k)$, $1 \le k \le d$ and unimodular lattice $\Lambda$, let $\hat{g}^k(\Lambda):=\sum_{(v_1,\dots,v_k)\in \Lambda^k \setminus 0} g(v_1,\dots, v_k)$. I speculate there might be such a generalization of Siegel's mean value formula:

$$\int_X \hat{g}^k d\mu = \int_{\mathbb R^d} \dots \int_{\mathbb R^d} g(v_1,\cdots,v_k) dv_1 \cdots dv_k$$

Is this correct and proved somewhere in the literature? Or can anyone suggest a proof for it below if not too complicated from the classical Siegel's formula?

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Such a generalization (roughly) exists, known as Rodger's Integration Formula. See Section 1.2 of Seungki Kim's Dissertation for a reference. Theorems 1.2 and 1.3 are of interest.

Theorem 1.2: (Rogers). Let $k < n$, and $\rho : (\mathbb{R}^n)^kk \to \mathbb{R}$ be a compactly supported, bounded, and Borel measurable function. Then $$ \int_{X}\sum_{\substack{v_1,\dots,v_k\in L\setminus\{0\}\\\mathsf{rank}(\langle v_1,\dots,v_k\rangle) = k}}\rho(v_1,\dots,v_k)d\mu = \int_{\mathbb{R}^n}\dots \int_{\mathbb{R}^n}\rho(x_1,\dots,x_k)dx_1\dots dx_k. $$

Up to the requirement that one sums over rank $k$ collections of vectors $(v_1,\dots,v_k)$, this is what you want. One can remove this restriction (to get the generalization that you want), but the formula becomes more complex --- there is an additional (additive) term on the right-hand side. See Theorem 1.3 of the dissertation for details.

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  • $\begingroup$ Thanks for this great reference! By the way although theorem 1.3 was proved by Rogers, the theorem 1.2 was not proved only in the German paper [17] by Schmidt in this thesis? Are there any other resources from which one could find a proof for theorem 1.2? $\endgroup$
    – taylor
    Oct 31, 2022 at 0:21
  • $\begingroup$ There's recently a proof of an adelic version of theorem 1.2, which you can likely extract the proof you want out of. Kim claims here that Rodgers' original proof contains an error, though apparently it can be repaired. $\endgroup$
    – Mark
    Oct 31, 2022 at 4:11

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