Oppenheim conjecture for pairs consisting of a linear form and a quadratic form

Unipotent Flows And Isotropic Quadratic Forms In Positive Characteristic

The analogous statement to Oppenheim conjecture over a local field of positive characteristic is proved. The dynamical argument is most involved in the case of characteristic 3.

Quadratic forms of signature $(2, 2)$ or $(3, 1)$ I: effective equidistribution in quotients of $\mathrm{SL}_4(\mathbb{R})$

We prove an effective equidistribution theorem for orbits of horospherical subgroups of $\mathrm{SO}(2, 2)$ and $\mathrm{SO}(3, 1)$ in quotients of $\mathrm{SL}_4(\mathbb{R})$ with a polynomial error term. In a forthcoming paper, we will use this theorem to prove an effective version of the Oppenheim conjecture for indefinite quadratic forms of signature $(2, 2)$ or $(3, 1)$ with a polynomial error rate.

Effective density for inhomogeneous quadratic forms II: fixed forms and generic shifts

We establish effective versions of Oppenheim's conjecture for generic inhomogeneous quadratic forms. We prove such results for fixed quadratic forms and generic shifts. Our results complement our companion paper where we considered generic forms and fixed shifts. In this paper, we use ergodic theorems and in particular we establish a strong spectral gap with effective bounds for some representations of orthogonal groups which do not possess Kazhdan's property (T).

秩2齐性空间中的有效等分布与二次型的取值

Effective equidistribution in rank 2 homogeneous spaces and values of quadratic forms

本文在绝对秩2的拟分裂几乎单线性代数群商空间中建立了带多项式误差率的幂幺轨道有效等分布定理，并据此给出不定三元二次型整点取值分布的定量结果，从而有效且定量地证明了Oppenheim猜想。

An effective equidistribution result for $SL(2,R)\ltimes(R^2)^{\oplus k}$ and application to inhomogeneous quadratic forms

Let $G=$SL$(2,R)\ltimes(R^2)^{\oplus k}$ and let $Γ$ be a congruence subgroup of SL$(2,Z)\ltimes(Z^2)^{\oplus k}$. We prove a polynomially effective asymptotic equidistribution result for special types of unipotent orbits in $Γ\backslash G$ which project to pieces of closed horocycles in SL$(2,Z)\backslash$SL$(2,R)$. As an application, we prove an effective quantitative Oppenheim type result for the quadratic form $(m_1-α)^2+(m_2-β)^2-(m_3-α)^2-(m_4-β)^2$, for $(α,β)$ of Diophantine type, following the approach by Marklof [24] using theta sums.

关于非齐次二次型的一个综述与一个结果

A survey and a result on inhomogeneous quadratic forms

本文综述无理非齐次二次型整数点取值的奥本海姆猜想类比研究，并证明二元情形下避开给定可数集的二次型集合具有满豪斯多夫维数，同时借助齐性动力学方法处理平移量固定的精细变体。

Pair correlation densities of inhomogeneous quadratic forms

Under explicit diophantine conditions on $(α,β)\in\RR^2$, we prove that the local two-point correlations of the sequence given by the values $(m-α)^2+\break (n-β)^2$, with $(m,n)\in\ZZ^2$, are those of a Poisson process. This partly confirms a conjecture of Berry and Tabor [2] on spectral statistics of quantized integrable systems, and also establishes a particular case of the quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms of signature (2,2). The proof uses theta sums and Ratner's classification of measures invariant under unipotent flows.

Values of Inhomogeneous Forms at S-integral points

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 form and the shift are allowed to vary. In order to do so, we prove analogues of Rogers' moment formulae for $S$-arithmetic congruence quotients as well as for the space of affine lattices. We believe the latter results to be of independent interest.

Margulis函数的矩与不定三元二次型

Moments of Margulis functions and indefinite ternary quadratic forms

本文通过建立Margulis α-函数沿单幂轨道扩张平移的λ-矩一致界，并引入新高度函数刻画经典Margulis不等式的失效，从而证明了不定三元二次型Oppenheim猜想的定量版本。

Values of quadratic forms on quasicrystals and a related problem

In this paper we study the set of values of quadratic form at points of a cut and project set. We will establish conditions which ensure that the set of values is dense. Our methods involve homogeneous dynamics and we will prove a orbit closure classification type result in this setting. This result has additional applications. In particular, we use it to study the set of integral values of a system consisting of a quadratic form and several linear forms.

整向量处线性型与二次型对的渐近分布

Asymptotic distribution for pairs of linear and quadratic forms at integral vectors

本文研究整向量上二次型与线性型对的值的联合分布问题，在特定维数、无理性及符号差条件下，得到满足范数与取值区间约束的整向量数量的渐近公式，推进了相关数论与动力系统领域的定量研究。

Dynamical system of a quadratic stochastic operator with two discontinuity points

In this paper we consider a population consisting of two species, dynamics of which is defined by a quadratic stochastic operator with variable coefficients, making it discontinuous operator at two points. This operator depends on three parameters. It is shown that under suitable conditions on the parameters this operator may have fixed points, convergence of trajectories and there may exist periodic points.

三次二次型在整数上的取值与3维扭体的伯里-塔布猜想

Values of ternary quadratic forms at integers and the Berry-Tabor conjecture for 3-tori

本文通过研究三次二次型在整数上的取值，验证了伯里-塔布猜想在三维扭体中的适用性，解决了特殊对角型在不同缩放下的分布问题。