A Mean Value Theorem in Geometry of Numbers

  title={A Mean Value Theorem in Geometry of Numbers},
  author={Carl Ludwig Siegel},
  journal={Annals of Mathematics},
  • C. L. Siegel
  • Published 1 April 1945
  • Mathematics
  • Annals of Mathematics

A dynamical Borel–Cantelli lemma via improvements to Dirichlet’s theorem

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

On a mean value formula for multiple sums over a lattice and its dual

. 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

Adelic Rogers integral formula

. 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

Lattice deformations in the Heisenberg group

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

Mean Value of $S$-arithmetic Siegel transform: Rogers' mean value theorem for $S$-arithmetic Siegel transform and applications to the geometry of numbers

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

Ultrametric logarithm laws, II

We prove positive characteristic versions of the logarithm laws of Sullivan and Kleinbock–Margulis and obtain related results in metric diophantine approximation.

Masstheorie in der Geometrie der Zahlen

List Decoding Random Euclidean Codes and Infinite Constellations

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.

List Decoding Random Euclidean Codes and Infinite Constellations

  • Yihan ZhangS. Vatedka
  • 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.