Successive minima of projective toric varieties

Some measure-theoretical and dynamical properties of successive minima on the space of unimodular lattices

In this article, we study the relation between lattice basis and successive minima and give an estimate for the measure-theoretical distribution of successive minima. As consequences, we also discuss some logarithm laws associated to higher sucessive minima.

环面簇子簇上环面度量的非阿基米德Monge-Ampère测度

Non-archimedean Monge-Ampère measures for toric metrics on subvarieties of toric varieties

本文研究环面簇子簇上环面度量的非阿基米德Monge-Ampère测度，利用热带芽与热带格林函数显式描述其局部结构，推广热带相交积，为阿贝尔簇情形提供环面类比。

Arithmetically equivalent number fields have approximately the same successive minima

Let $K$ and $K'$ be arithmetically equivalent number fields, both of degree $d$. We prove that $K$ and $K'$ have the same successive minima, up to a constant depending only on $d$. We give examples showing that one cannot expect equality.

A discrete analogue for Minkowski's second theorem on successive minima

The main result of this paper is an inequality relating the lattice point enumerator of a 3-dimensional, 0-symmetric convex body and its successive minima. This is an example of generalization of Minkowski's theorems on successive minima, where the volume is replaced by the discrete analogue, the lattice point enumerator. This problem is still open in higher dimensions, however, we introduce a stronger conjecture that shows a possibility of proof by induction on the dimension.

环面簇上的有理逼近

Rational approximations on toric varieties

本研究利用Salberger的万有挠子方法研究分裂环面簇上的有理点逼近问题，证明特定几何假设下最优逼近可由过不动点的极小次数有理曲线实现，证实McKinnon猜想，且这些曲线对应扇的中心本原集合。

Successive Minima and Best Simultaneous Diophantine Approximations

We study the problem of best approximations of a vector $α\in{\mathbb R}^n$ by rational vectors of a lattice $Λ\subset {\mathbb R}^n$ whose common denominator is bounded. To this end we introduce successive minima for a periodic lattice structure and extend some classical results from geometry of numbers to this structure. This leads to bounds for the best approximation problem which generalize and improve former results.

Lattice Points in Large Borel Sets and Successive Minima

Let $B$ be a Borel set in $\mathbb E^{d}$ with volume $V(B)=\infty$. It is shown that almost all lattices $L$ in $\mathbb E^{d}$ contain infinitely many pairwise disjoint $d$-tuples, that is sets of $d$ linearly independent points in $B$. A consequence of this result is the following: let $S$ be a star body in $\mathbb E^{d}$ with $V(S)=\infty$. Then for almost all lattices $L$ in $\mathbb E^{d}$ the successive minima $λ_{1}(S,L),..., λ_{d}(S,L)$ of $S$ with respect to $L$ are 0. A corresponding result holds for most lattices in the Baire category sense. A tool for the latter result is the sem…

尖点形式上 Petersson 范数的逐次最小值分布

Distribution of the successive minima of the Petersson norm on cusp forms

本文研究有限指数子群对应的整尖点形式格在 Petersson 范数下的逐次最小值渐近分布，并将主模群情形的已有结果推广到一般 level。

超椭圆函数域格的相继极小值、行列式与自同构群

Successive Minima, Determinant and Automorphism Groups of Hyperelliptic Function Field Lattices

该研究针对超椭圆函数域格，探究其相继极小值与行列式，还揭示了代数函数域自同构群和函数域格自同构群的关联，补充了函数域格领域的已有成果。

Fractional dimension related to badly approximable matrices associated with higher successive minima

In this article we introduce the notion of badly approximable matrices of higher order using higher sucessive minima in $\mathbb R^d$. We prove that for order less than $d$, they have Lebesgue measure zero and the gaps between them still have full Hausdorff dimension.

Low degree hypersurfaces of projective toric varieties defined over a $C_1$ field have a rational point

Quasi algebraically closed fields, or $C_1$ fields, are defined in terms of a low degree condition. Namely, the field $K$ is $C_1$ if every degree $d$ hypersurface of the projective space $\mathbb{P}_K^n$ contains a $K$-point as soon as $d\leq n$. In this article we define a notion of low toric degree generalizing this condition for hypersurfaces of simplicial projective split toric varieties. This allows us to prove a particular case of the $C_1$ conjecture of Kollár, Lang and Manin : any smooth separably rationally connected variety that can be embedded as such a hypersurface over a $C_1$ fi…

Reduction Theory of Algebraic Modules and their Successive Minima

Lattices defined as modules over algebraic rings or orders have garnered interest recently, particularly in the fields of cryptography and coding theory. Whilst there exist many attempts to generalise the conditions for LLL reduction to such lattices, there do not seem to be any attempts so far to generalise stronger notions of reduction such as Minkowski, HKZ and BKZ reduction. Moreover, most lattice reduction methods for modules over algebraic rings involve applying traditional techniques to the embedding of the module into real space, which distorts the structure of the algebra. In this pap…

Proof of W.M.Schmidt's conjecture concerning successive minima of a lattice

For a real $N\ge 1$ and a vector $ξ=(1,ξ_1,...,ξ_n)$ define a matrix $$ {\cal A} (ξ, N) = ({array}{ccccc} N^{-1} & 0& 0& ... &0 \cr N^{\frac{1}{n}} ξ_1 & -N^{\frac{1}{n}} & 0&... & 0 \cr N^{\frac{1}{n}} ξ_2 &0& -N^{\frac{1}{n}} & ... & 0 \cr ... &... &... &... \cr N^{\frac{1}{n}} ξ_n &0&0&... &- N^{\frac{1}{n}} {array}) $$ and a lattice $$ Λ(ξ, N) = {\cal A} (ξ, N)\mathbb{Z}^{n+1}. $$ Consider a convex 0-symmetric body $${\cal W} = \{z= (x,y_1,...,y_n)\in \mathbb{R}^{n+1}: \max (|x|, |y|)\le 1 \} >.$$ For a natural $l, 1\le l \le n+1$ let $μ_l (ξ, N)$ be the $l$-th successive minimum of ${\cal…

Minkowski bases, Korkin-Zolotarev bases and successive minima

Let $λ_k$ denote the $k$-th successive minimum of a lattice $L$. We study properties of the lengths of certain bases of $L$. If $v_1, \dots v_n$ is a basis which is reduced in the sense of Minkowski we show that $\lvert v_k \rvert^2 \leq \frac{k}{4} λ_{k}^2$ for $k = 6, 7$, confirming a conjecture of Schürmann, and obtaining the first improvement of a classical bound by Van der Waerden. We construct a sequences of lattices where $\lvert v_n \rvert$ is significantly longer than the longest vector in a Korkin-Zolotarev reduced basis, answering a question of Schürmann. In an appendix joint with L…