Related papers: Rogers' mean value theorem for $S$-arithmetic Sieg…

We develop the theory and properties of primitive unimodular $S$-arithmetic lattices in $\mathbb{Q}_S^d$ by giving integral formulas in the spirit of Siegel's primitive mean value formula and Rogers' and Schmidt's second moment formulas.…

Using results from spectral theory of Eisenstein series, we prove a formula for the second moment of the Siegel transform when averaged over the subspace of symplectic lattices. This generalizes the classical formula of Rogers for the…

We prove effective versions of Oppenheim's conjecture for generic inhomogeneous forms in the S-arithmetic setting. We prove an effective result for fixed rational shifts and generic forms and we also prove a result where both the quadratic…

Using classical results of Rogers bounding the $L^2$-norm of Siegel transforms, we give bounds on the heights of approximate integral solutions of quadratic equations and error terms in the quantiative Oppenheim theorem of…

We prove a uniform effective density theorem as well as an effective counting result for a generic system comprising a polynomial with a mild homogeneous condition and several linear forms using Roger's second moment formula for the Siegel…

We compute higher moments of the Siegel--Veech transform over quotients of $SL(2,\mathbb{R})$ by the Hecke triangle groups. After fixing a normalization of the Haar measure on $SL(2,\mathbb{R})$ we use geometric results and linear algebra…

We compute the first and second moment formulas for Siegel transforms related to problems counting primitive lattice points in the real plane with congruence conditions. As applications, we derive an analog of Schmidt's random counting…

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 application, we prove that for $L$…

We investigate lattice-counting problems associated with symplectic forms from the perspective of homogeneous dynamics. In the qualitative direction, we establish an analog of Margulis theorem for symplectic forms, proving density results…

We consider the question of how approximations satisfying Dirichlet's theorem spiral around vectors in $\mathbb{R}^d$. We give pointwise almost everywhere results (using only the Birkhoff ergodic theorem on the space of lattices). In…

In these notes, we refine Mitsui's Prime Number Theorem from 1957, which for a number field $K$ predicts how many prime elements there are in bounded convex sets in $K \otimes_{\mathbf Q} \mathbf R$, by incorporating potential Siegel zeros…

We study the light-cone Siegel transform, transforming functions on the light cone of a rational indefinite quadratic form $Q$ to a function on the homogenous space $\text{SO}^+_Q(\mathbb{Z})\backslash \text{SO}^+_Q(\mathbb{R})$. In…

In this paper, we prove a quantitative version of the Oppenheim conjecture for indefinite ternary quadratic forms: for any indefinite irrational ternary quadratic form $Q$ that is not extremely well approxiable by rational forms, and for…

We prove functional limit theorems for lattice point counting for affine and congruence lattices using the method of moments. Our main tools are higher moment formulae for Siegel transforms on the corresponding homogeneous spaces, which we…

This work has been motivated by recent papers that quantify the density of values of generic quadratic forms and other polynomials at integer points, in particular ones that use Rogers' second moment estimates. In this paper we establish…

Recently, Seungki Kim proved an extension of Rogers' mean value formula to the adeles of an arbitrary number field. In this paper we give a new proof Kim's formula, and give a criterion ensuring convergence in this formula. We also discuss…

Siegel defined zeta functions associated with indefinite quadratic forms, and proved their analytic properties such as analytic continuations and functional equations. Coefficients of these zeta functions are called measures of…

We give a sharpened form of Siegel Lemma's w. r. t. the maximum norm. This implies a new lower bound on the greatest element of a sum-distinct set of positive integers (Erd\"os-Moser problem). The main tools are Minkowski's theorem on…

Let $n \in \mathbb{Z}_{\geq 3}$ be given. We prove Lebesgue-almost everywhere pointwise inversion formulae for the Siegel transforms in the geometry of numbers. These inversion formulae are quite general; for instance, they are valid for…

We introduce and study the Weingarten calculus for centered random permutation matrices in the symmetric group S_N. After presenting a formulation of the Weingarten calculus on the symmetric group, we derive a formula in the centered case,…