Rogers integral formula

Here I will collect some interesting versions of Rogers integration formula. Will keep expanding as I find more.

Number of moments of Siegel transform Homogeneous space Due to
All moments \(\mathrm{SL}_{n}(\mathbb{R})/\mathrm{SL}_{n}(\mathbb{Z})\) Original [1]
2nd moment \(\mathrm{Sp}(2n,\mathbb{R})/\mathrm{Sp}(2n,\mathbb{Z})\) [2]
2nd moment \(\mathrm{SL}_{n}(\mathbb{R})/\Gamma\) for some congruence subgroup \(\Gamma\) [3]
All moments \(\mathrm{SL}_{r}(K \otimes \mathbb{R})/\mathrm{SL}_{r}(\mathcal{O}_{K})\) for a number field \(K\) [4]
All moments Affine space of lattices [5]

References

1.
C. A. Rogers, The number of lattice points in a set. Proceedings of the London Mathematical Society, 3 (1956) 305–320.
2.
D. Kelmer & S. Yu, The second moment of the siegel transform in the space of symplectic lattices. International Mathematics Research Notices, 2021 (2021) 5825–5859.
3.
A. Ghosh, D. Kelmer, & S. Yu, Effective density for inhomogeneous quadratic forms i: Generic forms and fixed shifts. International Mathematics Research Notices, 2022 (2022) 4682–4719.
4.
N. P. Gargava, V. Serban, & M. Viazovska, Moments of the number of points in a bounded set for number field lattices. in preparation, (2023).
5.
M. Alam, A. Ghosh, & J. Han, Higher moment formulae and limiting distributions of lattice points. Journal of the Institute of Mathematics of Jussieu, 23 (2024) 2081–2125.

This page was updated on February 10, 2026.
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