## 225 Citations

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

- MathematicsMoscow 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…

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

- 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…

### Adelic Rogers integral formula

- Mathematics
- 2022

. 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

- MathematicsGroups, 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…

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

- Mathematics
- 2019

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

- Mathematics
- 2011

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

### List Decoding Random Euclidean Codes and Infinite Constellations

- Computer ScienceIEEE 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.

### List Decoding Random Euclidean Codes and Infinite Constellations

- Computer Science2019 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.