Exponents of Diophantine approximation
Yann Bugeaud, Michel Laurent
本文主要综述了丢番图逼近的最新结果，着重讨论了各种丢番图逼近概念的指数度量的性质。
The paper is mostly a survey on recent results in Diophantine approximation, with emphasis on properties of exponents measuring various notions of Diophantine <approximation.
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Multiplicative Diophantine Exponents of Hyperplanes
研究了向量的乘性丢番图逼近性质，并利用动力学方法计算了超平面的丢番图指数。
Diophantine methods for exponential sums, and exponential sums for Diophantine problems
讨论了Hardy-Littlewood方法在理论和应用方面的最新进展，重点讨论了对角丢番图问题。最近有效的差分方法估计平均值的指数和第一，专注于发展涉及光滑外尔和。其次，讨论了经典的贝塞尔不等式和柯西-施瓦茨不等式的算术变体。最后，一些新兴的圆的方法和算术几何之间的联系被提及。
On Diophantine exponents in dimension 4
得到了四个实数同时丢番图逼近的普通丢番图指数与一致丢番图指数之间的一些新的不等式.
Geometry of Diophantine exponents
本综述探讨了丢番图指数及其满足的关系，介绍了经典丢番图指数、带权与乘性类似物及格的指数，并重点关注转换原理。
In this paper we give a survey of what is currently known about Diophantine exponents of lattices and propose several problems.
We survey classical and recent results on exponents of Diophantine approximation. We give only a few proofs and highlight several open problems.
On the spectrum of Diophantine exponents of lattices
本文研究了任意维数格点的弱均匀丢番图指数的谱特性。
Diophantine Exponents of Affine Subspaces: The Simultaneous Approximation Case
我们应用齐性空间上流的非发散估计来计算$\R^n$的仿射子空间及其非退化子流形的丢番图指数。
In this paper we define Diophantine exponents of lattices and investigate some of their properties. We prove transference inequalities and construct some examples with the help of Schmidt's subspace theorem.
Multiplicative Diophantine exponents of hyperplanes and their nondegenerate submanifolds
我们考虑单分子格空间上的多参数动力学。随着定量的非发散，我们证明了乘法丢番图指数的超平面是继承其非退化子流形。
On uniform Diophantine exponents of lattices
本文研究格的弱均匀丢番图指数谱，并在二维情形下给出了该谱的完整刻画。
In this paper we prove an existence theorem concerning linear forms of a given Diophantine type and apply it to study the structure of the spectrum of lattice exponents.
On Diophantine exponents of lattices
本文研究了d维格点的丢番图指数，通过归约到二维情况并应用度量理论方法，揭示了其指数谱特性。
On the weak uniform Diophantine exponent of a real number
本文提出实数的弱一致丢番图指数新概念，并完整刻画了该指数的所有可能取值谱。
Exponents of inhomogeneous Diophantine Approximation
在丢番图逼近中，非齐次问题通过所谓的迁移定理与齐次问题联系起来。我们通过引入新的丢番图逼近指数来重新审视这个经典话题。证明了n个线性型系统对R^n中一个泛点的逼近指数等于对偶型系统的一致齐次指数的倒数。
On triviality of uniform Diophantine exponents of lattices
本文研究格的一致丢番图指数，证明其仅能取得平凡值。
Intermediate Diophantine exponents and parametric geometry of numbers
这是对论文arXiv：1105.1554和arXiv：1105.5823的修订汇编。我们发展了一些属于W.Schmidt和L.Summerer的想法来定义中间丢番图指数，并将几个迁移不等式分解成一个中间指数的不等式链。
Transference inequalities for multiplicative Diophantine exponents
在本文中，我们证明了丢番图指数的乘法类似物的不等式，类似于经典情况中已知的不等式。特别地，我们证明了一个矩阵是坏逼近的当且仅当它的转置是坏逼近的，并建立了一些不等式连接乘法指数与普通的。
On Diophantine exponents and Khintchine's transference principle
本文改进了Jarnik和Apfelbeck对线性型转置系统的一致丢番图指数的估计，并将Laurent和Bugeaud对个别指数的估计推广到任意系统的情形.该方法还给出了Mahler迁移定理中一个较好的常数。
An inequality for Diophantine exponents in the case of simultaneous approximation
本文在同时逼近的情形下证明了一个关于个别丢番图指数和一致丢番图指数的不等式。这个不等式比Jarnik的更好的一致指数的小值。
On a Problem in Diophantine Approximation
证明了丢番图逼近中Littlewood和Mixed Littlewood拓扑的新结果。
We study some problems in metric Diophantine approximation over local fields of positive characteristic.
Diophantine approximation on lines with prime constraints
研究了R^2中具有素分子和素分母的直线上的丢番图逼近问题。
On Harrap's conjecture in Diophantine approximation
证明了Stephen Harrap关于与${\rm BAD}(α,β)$集有关的非齐次线性丢番图逼近的一个猜想.
We provide a lower bound for the ratio between the ordinary and uniform exponent of both simultaneous Diophantine approximation and Diophantine approximation by linear forms in any dimension. This lower bound was conjectured by Schmidt and Summerer and already shown in dimension $2$ and $3$. This lower bound is reached at regular systems presented in the context of parametric geometry of numbers, and thus optimal.
Using the Parametric Geometry of Numbers introduced recently by W.M. Schmidt and L. Summerer and results by D. Roy, we establish that the spectrum of the $2n$ exponents of Diophantine approximation in dimension $n\geq3$ is a subset of $\mathbb{R}^{2n}$ with non empty interior.
Let $a$, $b$, $c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this survey, we consider some unsolved problems and related works concerning the positive integer solutions $(x,y,z)$ of the ternary purely exponential diophantine equation $a^x + b^y = c^z$.
The best Diophantine approximations: the phenomenon of degenerate dimension
本文主要讨论多维线性型的丢番图逼近和同时的丢番图逼近。讨论了由最佳丢番图逼近生成的线性子空间的退化维数现象。这些结果大多数是作者在几年前建立的。在这里，我们收集所有这些在一起，并给出一些新的配方。
Diophantine approximations with positive integers: some remarks
本文对W. M.关于正整数的丢番图逼近的Schmidt定理和我们最近的一些结果。
We study the problem of Diophantine approximation on lines in $\mathbb{R}^d$ under certain primality restrictions.
We study the problem of Diophantine approximation on lines in $\mathbb{C}^2$ with numerators and denominators restricted to Gaussian primes.
Construction of subspaces with known Diophantine exponents for the last angle
本文利用数的几何方法，构造了 $\R^n$ 中能够计算最后一个角度关联丢番图指数的线性子空间，推广了实数的无理性测度。
Simultaneous two-dimensional best Diophantine approximations in the Euclidean norm
证明了二维最佳丢番图逼近关于欧氏范数的增长指数的一个新的下界。
In this note, we present an improvement to a recent result due to Beresnevich, Levesley, and Ward (2021) pertaining to weighted simultaneous Diophantine approximation on manifolds.
Distribution modulo one and diophantine approximation by Yann Bugeaud
书评：Distribution modulo one and丢番图逼近，Yann Bugeaud著，剑桥大学出版社，2012年。ISBN 978-0521111690，316页。
Independence of the Diophantine exponents associated with linear subspaces
本文研究Schmidt提出的丢番图逼近推广问题，证明线性子空间丢番图指数相关函数间无光滑关系，并构造可计算指数的子空间。
Intersective polynomials and Diophantine approximation, II
应用Schmidt格方法，证明了在满足一定局部条件下，单素变量多项式系统模1的同时丢番图逼近的结果.
Weak approximations, Diophantine exponents and two-dimensional lattices
本文研究二维格点的迪奥菲安指数及其相关弱统一逼近性质，利用连分数理论分析普通与统一迪奥菲安指数间的不等式关系，揭示两个实数无理数度量函数的行为关联。
Best Simultaneous Diophantine Approximations under a Constraint on the Denominator
我们研究了Jurkat提出的在分母上有约束条件下的最佳同时丢番图逼近问题。新的较低的估计最优逼近常数的临界行列式的适当的恒星机构。工具是由具有较小分母的有理数对有理数的同时丢番图逼近的结果。最后，将近似结果应用于整数向量的分解。
We compute the sequence of best Diophantine approximations for some pairs of cubic Pisot numbers which do not satisfy the Property (F).
We establish an explicit asymptotic formula for the number of rational solutions of intrinsic Diophantine inequalities on simply-connected simple algebraic groups, at arbitrarily small scales.
In this paper we prove transference inequalities for regular and uniform Diophantine exponents in the weighted setting. Our results generalize the corresponding inequalities that exist in the `non-weighted' case.
In this paper we study the three-dimensional analogue of the relation between the irrationality exponent of a real number and the growth of its regular continued fraction partial quotients. As a multidimensional generalisation of continued fractions, we consider Klein polyhedra.
Diophantine approximation and automorphic spectrum
本文建立了半单代数群的齐次簇的丢番图逼近速度的定量估计。所建立的估计推广和改进了以前的估计，并且在许多情况下是尖锐的。我们表明，丢番图逼近的速度是由频谱的自守表示，因此受到广义拉马努金几何。
Exponents of Diophantine Approximation and Sturmian Continued Fractions
设x为实数，n为正整数。定义了丢番图逼近的四个指数，它们是Mahler和Koksma定义的指数w_n（x）和w_n^*（x）的补充.当n=2，x是一个实数，其连分数展开式与某些正整数的Sturmian序列一致时，我们计算了它们的六个值，直到初始项。特别地，我们得到了这样一个连分数x的二次根号逼近的精确指数。
Diophantine exponents for systems of linear forms in two variables
对两个整数变量的$ n\ge 2$实线性型组，改进了Jarn\'ık不等式中的一致丢番图指数$α$与普通丢番图指数$β$之间的不等式. Jarn 'ık（1949，1954）证明了$β\ge α(α-1)$ .在本文中，我们给出了$α>1$情形下的一个较好的上界.我们证明了β PDMATHAMR 1/2（α^2-α+1+ \sqrt{(α^2-α+1)^2 +4α^2(α-1)}）if 1 \le α \le 2 1/2（α^2-1+ \sqrt{(α^2-1)^2+4α(α-1)}）if α \ge 2
We introduce diophantine approximation groups and their associated Kronecker foliations, using them to provide new algebraic and geometric characterizations of $K$-linear and algebraic dependence. As a consequence we find reformulations -- as algebraic and geometric (graph) rigidities -- of the Theorems of Baker and Lindemann-Weierstrass, the Logarithm Conjecture and the Schanuel Conjecture. There is an Appendix describing diophantine approximation groups as model theoretic types.
On some open problems in Diophantine approximation
讨论了丢番图逼近中的几个公开问题。其中既有著名的Littlewood和Zaremba定理，也有一些新的不太著名的问题。
Theorem on best Diophantine approximations for linear forms
证明了关于线性型的最佳丢番图逼近所张成的子空间维数退化的一个新的定量结果。
In these notes, we begin by recalling aspects of the classical theory of metric Diophantine approximation; such as theorems of Khintchine, Jarn\'ık, Duffin-Schaeffer and Gallagher. We then describe recent strengthening of various classical statements as well as recent developments in the area of Diophantine approximation on manifolds. The latter includes the well approximable, the badly approximable and the inhomogeneous aspects.
Metric theory of inhomogeneous Diophantine approximations with a fixed matrix
本文针对固定矩阵情形，利用转移原理建立矩阵向量配对与转置矩阵丢番图性质的双向关联，证明齐次丢番图逼近经典结论的对应版本，解答相关公开问题，构建了对应的非齐次丢番图逼近度量理论。
Diophantine approximation with arithmetic functions, I
本文证明了一个强的同时丢番图逼近定理，它适用于加法函数和乘性函数的值，只要这两类函数在素数上具有一定的正则性。
Analogues of the classical theorems of Khintchine, Jarnik and Jarnik-Besicovitch in the metrical theory of Diophantine approximation are established for quaternions by applying results on the measure of general `lim sup' sets.
Zero-one laws in simultaneous and multiplicative Diophantine approximation
讨论Beresnevich和Velani提出的两个问题，得到了同时和乘性丢番图逼近的0 - 1定律。我们的证明依赖于一个Cassels-Gallagher型定理，以及一个更高的维度模拟的交叉重叠原则的Beresnevich，Haynes和Velani。
On Diophantine approximations with positive integers: a remark to W.M.Schmidt's theorem
我们证明了W.M.施密特定理与$x_1,x_2$为{ \it正}整数的$α_1x_1+α_2x_2 +y$型线性型的丢番图逼近有关。
We extend the Khintchine transference inequalities, as well as a homogeneous-inhomogeneous transference inequality for lattices, due to Bugeaud and Laurent, to a weighted setting. We also provide applications to inhomogeneous Diophantine approximation on manifolds and to weighted badly approximable vectors. Finally, we interpret and prove a conjecture of Beresnevich-Velani (2010) about inhomogeneous intermediate exponents.
Diophantine approximation exponents on homogeneous varieties
近年来，在丢番图逼近和齐次动力学的界面上取得了非常重要的进展。在本文的第一部分中，我们给出了一个简短的阐述字典开发的Dani和Kleinbock-Margulis的关系丢番图性质的向量分布的轨道流的空间上的幺模格。在本文的第二部分中，我们简要地描述了作者最近开发的这本字典的扩展，它建立了一个类似的动力学对应一般格轨道上的齐性空间。我们专注于具体的问题估计指数的丢番图逼近算术格作用于代数簇。在第三部分中，我们通过在一些基本情况下建立稠格轨道的丢番图指数的显式界，来证明我们的结果。其中包括对仿射空间的线性作用和仿射作用，以及对固定行列式矩阵簇的作用。在某些情况下，这些指数被证明是最好的可能。
Metric Diophantine approximation: The Khintchine--Groshev theorem for non-degenerate manifolds
本文的主要目的是证明非退化流形上线性丢番图逼近的一个Khintchine型发散定理，完善了以前关于收敛性的结果.
Some remarks on diophantine equations and diophantine approximation
我们首先回忆一下这种联系，回到A。因此，代数数的有理逼近与某些丢番图方程的整数解之间存在着联系。其次，我们回顾几个有限性结果之间的等价性各种丢番图方程。我们也给出了许多等价的声明马勒的推广的基本定理的图厄。特别是，我们表明，度$3$的Thue-马勒定理意味着定理的Thue-马勒任意度$\ge3$，我们与西格尔定理的合理的积分点的投影线$¶^1(K)$减去$3$点。最后，我们将我们的研究扩展到高维空间与施密特的子空间定理。
We prove that the matrix analogue of the Veronese curve is strongly extremal in the sense of Diophantine approximation, thereby resolving a question posed by Beresnevich, Kleinbock, and Margulis ('15) in the affirmative.
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