A generic effective Oppenheim theorem for systems of forms

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

Let Q be a nondegenerate quadratic form, and L is a nonzero linear form of dimension d>3. As a generalization of the Oppenheim conjecture, we prove that the set {(Q(x),L(x)):x\in Z^d} is dense in R^2 provided that Q and L satisfy some natural conditions. The proof uses dynamics on homogeneous spaces of Lie groups.

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

We prove higher moment formulas for Siegel transforms defined over the space of unimodular $S$-lattices in $\mathbb Q_S^d$, $d\ge 3$, where in the real case, the formulas are introduced by Rogers (1955). As applications, we obtain the random statements of Gauss circle problem for any convex sets in $\mathbb Q_S^d$ containing the origin and of the effective Oppenheim conjecture for $S$-arithmetic quadratic forms.

On Oppenheim-type conjecture for systems of quadratic forms

Let Q_i, i=1,...,t, be real nondegenerate indefinite quadratic forms in d variables. We investigate under what conditions the closure of the set {(Q_1(x),...,Q_t(x)): x\in Z^d-{0}} contains (0,..,0). As a corollary, we deduce several results on the magnitude of the set Δof g\in GL(d,R) such that the closure of the set {(Q_1(gx),...,Q_t(gx)): x\in Z^d-{0}} contains (0,...,0). Special cases are described when depending on the mutual position of the hypersurfaces {Q_i=0}, i=1,...,t, the set Δhas full Haar measure or measure zero and Hausdorff dimension d^2-(d-2)/2.

随机二次型的定量Oppenheim猜想与函数域中的最优方差界

Quantitative Oppenheim Conjecture for Random Quadratic Forms and Optimal Variance Bounds in Function Fields

在函数域中证明了Oppenheim猜想的定量版本，通过计算Siegel变换的高阶矩，得到了集合中格点数的方差最优界，并精确计算了球内格点数的方差。

Explicit solutions to the Oppenheim conjecture for indefinite ternary diagonal forms

We prove the Oppenheim conjecture for indefinite ternary diagonal forms of the type $x^{2}+y^{2} -αz^{2}$ where $ α$ is an irrational number. Our method is explicit in the sense that we are able to construct a solution to the problem and we obtain an effective bound on the solution. The method is geometrical and is based on continued fractions.

Quantitative Oppenheim Conjecture for Quadratic Forms in 5 Variables over Function Fields

We translate Davenport's and Heilbronn's work on a quantitative version of the Oppenheim conjecture for indefinite diagonal quadratic forms in 5 variables into the setting of function fields.

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).

On the density at integer points of a system comprising an inhomogeneous quadratic form and a linear form

We prove an analogue of the Oppenheim conjecture for a system comprising an inhomogeneous quadratic form and a linear form in $3$ variables using dynamics on the space of affine lattices.

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。

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.

多项式有效等分布对于某些单位元子群在完美李群商中的应用

Polynomially effective equidistribution for certain unipotent subgroups in quotients of perfect Lie groups

本文证明了完美李群商中某些单位元子群轨道的有效等分布定理，并应用该结果得到不定二次型的有效Oppenheim猜想版本。

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.