On the distribution of non-primitive lattice points in the plane

大维数下本原格点的分布

Distribution of Primitive Lattice Points in Large Dimensions

本文利用本原 Siegel 变换的高阶矩公式，研究高维对称 Borel 集中本原格点的分布，证明固定体积下的 Poisson 极限与次指数增长体积下的正态极限，并推广至随机过程。

Single-Class Genera of Positive Integral Lattices

We give an enumeration of all positive definite primitive Z-lattices in dimension >= 3 whose genus consists of a single isometry class. This is achieved by using bounds obtained from the Smith-Minkowski-Siegel mass formula to computationally construct the square-free determinant lattices with this property, and then repeatedly calculating pre-images under a mapping first introduced by G. L. Watson. We hereby complete the classification of single-class genera in dimensions 4 and 5 and correct some mistakes in Watson's classifications in other dimensions. A list of all single-class primitive Z-l…

Primitive point packing

A point in the $d$-dimensional integer lattice $\mathbb{Z}^d$ is primitive when its coordinates are relatively prime. Two primitive points are multiples of one another when they are opposite, and for this reason, we consider half of the primitive points within the lattice, the ones whose first non-zero coordinate is positive. We solve the packing problem that asks for the largest possible number of such points whose absolute values of any given coordinate sum to at most a fixed integer $k$. We present several consequences of this result at the intersection of geometry, number theory, and combi…

S-算术本原 Siegel 变换的均值定理

Mean value theorems for the S-arithmetic primitive Siegel transforms

本文发展了本原幺模 $S$-算术格的理论，给出了 $S$-本原 Siegel 变换的均值与二阶矩公式，并将其应用于格点计数、定量 Khintchine--Groshev 定理及幂流的 $S$-算术对数律。

On Kronecker's density theorem, primitive points and orbits of matrices

We discuss recent quantitative results in connexion with Kronecker's theorem on the density of subgroups in R^n and with Dani and Raghavan's theorem on the density of orbits in the spaces of frames. We also propose several related problems. The case of the natural linear action of the unimodular group SL_2(Z) on the real plane is investigated more closely. We then establish an intriguing link between the configuration of (discrete) orbits of primitive points and the rate of density of dense orbits.

ADC二次格在代数数域上的局部信息

Local information of ADC quadratic lattices over algebraic number fields

本文研究了ADC二次格在代数数域上的局部结构，确定了二元和三元格的表示集、计数公式及密度，并探讨了其在非阿基米德局部域上的性质与应用。

正特征下格点的有效等分布

Effective equidistribution of lattice points in positive characteristic

该研究针对有限域上整体函数域的位对应空间，利用Goronik和Nevo的格点计数技术，得到了正特征下本原格点及gcd方程重整化解的有效等分布结论，是整数格点相关结果在正特征下的更精确类比。

Primitive points in rational polygons

Let $\mathcal A$ be a star-shaped polygon in the plane, with rational vertices, containing the origin. The number of primitive lattice points in the dilate $t\mathcal A$ is asymptotically $\frac6{π^2}$ Area$(t\mathcal A)$ as $t\to \infty$. We show that the error term is both $Ω_\pm\big( t\sqrt{\log\log t} \big)$ and $O(t(\log t)^{2/3}(\log\log t)^{4/3})$. Both bounds extend (to the above class of polygons) known results for the isosceles right triangle, which appear in the literature as bounds for the error term in the summatory function for Euler's $ϕ(n)$.

丢番图规避与数域理想中的小高度本原元

Diophantine avoidance and small-height primitive elements in ideals of number fields

针对数域理想中的本原元问题，结合丢番图规避结果推导格点小范数点结论，给出二次域理想本原元最小高度明确界、全实数域全正本原元及生成多项式Mahler测度等相关界。

$p$-adic Directions of Primitive Vectors

Linnik type problems concern the distribution of projections of integral points on the unit sphere as their norm increases, and different generalizations of this phenomenon. Our work addresses a question of this type: we prove the uniform distribution of the projections of primitive $\mathbb{Z}^{2}$ points in the $p$-adic unit sphere, as their (real) norm tends to infinity. The proof is via counting lattice points in semi-simple $S$-arithmetic groups.

Principal factors and lattice minima

Let $\mathit{k}=\mathbb{Q}(\sqrt[3]{d},ζ_3)$, where $d>1$ is a cube-free positive integer, $\mathit{k}_0=\mathbb{Q}(ζ_3)$ be the cyclotomic field containing a primitive cube root of unity $ζ_3$, and $G=\operatorname{Gal}(\mathit{k}/\mathit{k}_0)$. The possible prime factorizations of $d$ in our main result [2, Thm. 1.1] give rise to new phenomena concerning the chain $Θ=(θ_i)_{i\in\mathbb{Z}}$ of \textit{lattice minima} in the underlying pure cubic subfield $L=\mathbb{Q}(\sqrt[3]{d})$ of $\mathit{k}$. The aims of the present work are to give criteria for the occurrence of generators of primiti…

Distribution of Shapes of orthogonal Lattices

It was recently shown by Aka, Einsiedler and Shapira that if d>2, the set of primitive vectors on large spheres when projected to the d-1-dimensional sphere coupled with the shape of the lattice in their orthogonal complement equidistribute in the product space of the sphere with the space of shapes of d-1-dimensional lattices. Specifically, for d=3,4,5 some congruence conditions are assumed. By using recent advances in the theory of unipotent flows, we effectivize the dynamical proof to remove those conditions for d=4,5. It also follows that equidistribution takes place with a polynomial erro…

具有 $k\ge3$ 个内部点的凸格点多边形

Convex Lattice Polygons with $k\ge3$ Interior Points

本文研究了具有 $n$ 个边界格点和 $k\geq 3$ 个共线内部格点的凸格 $n$ 边形的几何性质，提出了一种通过构造本原格三角形来添加边的方法，并给出了必要条件与次数上界，最终完整分类了存在具有 $k$ 个共线及非共线内部点的凸 $n$ 边形的正整数。

Approximation of points in the plane by generic lattice orbits

We give upper and lower bounds for Diophantine exponents measuring how well a point in the plane can be approximated by points in the orbit of a lattice $Γ<\mathrm{SL}_2(\mathbb{R})$ acting linearly on $\mathbb{R}^2$. Our method gives bounds that are uniform for almost all orbits.

Some results on random unimodular lattices

Let $n \in \mathbb{Z}_{\geq 3}.$ Given any Borel subset $A$ of $\mathbb{R}^n$ with finite and nonzero measure, we prove that the probability that the set of primitive points of a random full-rank unimodular lattice in $\mathbb{R}^n$ does not contain any $\mathbb{R}$-linearly independent subset of $A$ of cardinality $(n-2)$ is bounded from above by a constant multiple, which depends only on $n$, of $\left(\mathrm{vol}(A)\right)^{-1}.$ This generalizes a result that is jointly due to J. S. Athreya and G. A. Margulis (see \cite[Theorem 2.2]{Log}). We also generalize independent results of C. A. R…

On arithmetic lattices in the plane

We investigate similarity classes of arithmetic lattices in the plane. We introduce a natural height function on the set of such similarity classes, and give asymptotic estimates on the number of all arithmetic similarity classes, semi-stable arithmetic similarity classes, and well-rounded arithmetic similarity classes of bounded height as the bound tends to infinity. We also briefly discuss some properties of the $j$-invariant corresponding to similarity classes of planar lattices.

平面格中有理角的分类 II

Classification of rational angles in plane lattices II

本文延续先前工作，完成了平面格中形成π的有理倍数角的点配置分类，并给出了格子的完整明确描述。

双曲平面上二次格的Zeta函数

Zeta functions of quadratic lattices of a hyperbolic plane

本文研究了双曲平面上二次格满秩子格真等价类的Dirichlet级数，推导出相关Zeta函数的显式公式与组合计算方法，并揭示了大量真类为单格类的解析性质。

扩展极限球面上本原有理点的等分布

Equidistribution of primitive rational points on expanding horospheres

本文证实了Marklof关于扩展极限球面上稀疏算术点集等分布的猜想，并借此给出了Schmidt关于正交于整数向量的格形状分布结果的类比。

On Kemnitz' Conjecture Concerning Lattice Points in the Plane

In 1961, P. Erdős, A. Ginzburg, and A. Ziv proved a remarkable theorem stating that each set of $2n-1$ integers contains a subset of size $n$, the sum of whose elements is divisible by $n$. We will prove a similar result for pairs of integers, i.e., planar lattice points, usually referred to as Kemnitz' conjecture.

Effective joint equidistribution of primitive rational points on expanding horospheres

We prove an effective version of a result due to Einsiedler, Mozes, Shah and Shapira who established the equidistribution of primitive rational points on expanding horospheres in the space of unimodular lattices in at least $3$ dimensions. Their proof uses techniques from homogeneous dynamics and relies in particular on measure-classification theorems -- an approach which does not lend itself to effective bounds. We implement a strategy based on spectral theory, Fourier analysis and Weil's bound for Kloosterman sums in order to quantify the rate of equidistribution for a specific horospherical…

Weak admissibility, primitivity, o-minimality, and Diophantine approximation

We generalise M. M. Skriganov's notion of weak admissibility for lattices to include standard lattices occurring in Diophantine approximation and algebraic number theory, and we prove estimates for the number of lattice points in sets such as aligned boxes. Our result improves on Skriganov's celebrated counting result if the box is sufficiently distorted, the lattice is not admissible, and, e.g., symplectic or orthogonal. We establish a criterion under which our error term is sharp, and we provide examples in dimensions $2$ and $3$ using continued fractions. We also establish a similar countin…

The Probability of Choosing Primitive Sets

We generalize a theorem of Nymann that the density of points in Z^d that are visible from the origin is 1/zeta(d), where zeta(a) is the Riemann zeta function 1/1^a + 1/2^a + 1/3^a + ... A subset S of Z^d is called primitive if it is a Z-basis for the lattice composed of the integer points in the R-span of S, or, equivalently, if S can be completed to a Z-basis of Z^d. We prove that if m points in Z^d are chosen uniformly and independently at random from a large box, then as the size of the box goes to infinity, the probability that the points form a primitive set approaches 1/[ζ(d)ζ(d-1)...zet…

Equidistribution and freeness on Grassmannians

We associate a certain tensor product lattice to any primitive integer lattice and ask about its typical shape. These lattices are related to the tangent bundle of Grassmannians and their study is motivated by Peyre's programme on "freeness" for rational points of bounded height on Fano varieties.

Spherical equidistribution in adelic lattices and applications

In this paper we study spherical equidistribution on the space of (translates of) adelic lattices, which we apply to understand the fine-scale statistics of the directions in the set of shifted primitive lattice points. We also apply our results to the distribution of the free path lengths in the Boltzmann--Grad limit for point sets such as (possibly non-rational) translates of the lattice points all of whose coordinates are squarefree. Besides the equidistribution results for translates of expanding horospheres, a key ingredient is a probabilistic argument which allows us to tackle the techni…

Mean lattice point discrepancy bounds, II: Convex domains in the plane

We consider planar curved strictly convex domains with no or very weak smoothness assumptions and prove sharp bounds for square-functions associated to the lattice point discrepancy.

扩张 horosphere 上本原有理点的有效等分布

Effective equidistribution of primitive rational points on expanding horospheres

本文利用矩阵 Kloosterman 和界、Hecke 点有效等分布及 Rogers 积分公式，证明扩张闭 horosphere 上本原有理点的有效等分布，并给出线性同余小解与有限域非方阵计数应用。

An equivariant covering map from the upper half plane to the complex plane minus a lattice

This paper studies a covering map phi from the upper half plane to the complex plane with a triangular lattice excised. This map is interesting as it factorises Klein's J invariant. Its derivative has properties which are a slight generalisation of modular functions, and (phi')^6 is a modular function of weight 12. There is a homomorphism from the modular group Gamma to the affine transformations of the complex plane which preserve the excised lattice. With respect to this action phi is a map of Gamma-sets. Identification of the excised lattice with the root lattice of sl_3(C) allows functions…

平面格中有理角的分类

Classification of rational angles in plane lattices

本文研究平面格中可形成 π 的有理倍数角的点构型，通过允许格任意变化来分类这些角的数量与位置，并揭示其与正亏格代数曲线上有理点参数化的深刻联系。