# A Mean Value Theorem in Geometry of Numbers

```@article{Siegel1945AMV,
title={A Mean Value Theorem in Geometry of Numbers},
author={Carl Ludwig Siegel},
journal={Annals of Mathematics},
year={1945},
volume={46},
pages={340}
}```
• C. L. Siegel
• Published 1 April 1945
• Mathematics
• Annals of Mathematics
225 Citations
• Mathematics
Moscow Journal of Combinatorics and Number Theory
• 2020
Let \$X\cong \operatorname{SL}_2(\mathbb R)/\operatorname{SL}_2(\mathbb Z)\$ be the space of unimodular lattices in \$\mathbb R^2\$, and for any \$r\ge 0\$ denote by \$K_r\subset X\$ the set of lattices such
• Mathematics
• 2022
. We prove a generalized version of Rogers’ mean value formula in the space X n of unimodular lattices in R n , which gives the mean value of a multiple sum over a lattice L and its dual L ∗ . As an
. We formulate and prove the extension of the Rogers integral formula ([23]) to the adeles of number ﬁelds. We also prove the second moment formulas for a few important cases, enabling a number of
• Mathematics
Groups, Geometry, and Dynamics
• 2020
The space of deformations of the integer Heisenberg group under the action of \$\textrm{Aut}(H(\mathbb{R}))\$ is a homogeneous space for a non-reductive group. We analyze its structure as a measurable
We prove the second moment theorem for Siegel transform defined over the space of unimodular \$S\$-lattices in \$\mathbb Q_S^d\$, \$d\ge 3\$, following the work of Rogers (1955). As applications, we obtain
• Mathematics
• 2011
We prove positive characteristic versions of the logarithm laws of Sullivan and Kleinbock–Margulis and obtain related results in metric diophantine approximation.
• Computer Science
IEEE Transactions on Information Theory
• 2022
It is shown that if a certain conjecture that originates in analytic number theory is true, then the list size grows as a polynomial function of the gap-to-capacity.
• Computer Science
2019 IEEE International Symposium on Information Theory (ISIT)
• 2019
The goal of the present paper is to obtain a better understanding of the smallest achievable list size as a function of the gap to capacity, and shows a reduction from arbitrary codes to spherical codes, and derives a lower bound on the list size of typical random spherical codes.