Arnaud Durand
We investigate the large intersection properties of the set of points that are approximated at a certain rate by a family of affine subspaces. We then apply our results to various sets arising in the metric theory of Diophantine approximation, in the study of the homeomorphisms of the circle and in the perturbation theory for Hamiltonian systems.
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Slow polynomial mixing, dynamical Borel-Cantelli lemma and Hausdorff dimension of dynamical diophantine sets
本文针对缓慢的多项式混合系统，建立了动力学Borel-Cantelli引理与特定动力学丢番图集Hausdorff维数的最优性结果。
Hausdorff dimension of intersections between the Jarník sets and Diophantine fractals
研究了Jarník集与迪奥菲内分数集合交集的Hausdorff维度，证明其为满维，并指出无理数指数大于2且部分商有界的集合维度为1。
We place the theory of metric Diophantine approximation on manifolds into a broader context of studying Diophantine properties of points generic with respect to certain measures on $\Bbb R^n$. The correspondence between multidimensional Diophantine approximation and dynamics of lattices in Euclidean spaces is discussed in an elementary way, and several recent results obtained by means of this correspondence are surveyed.
We consider classes $\mathscr{G}^s ([0,1])$ of subsets of $[0,1]$, originally introduced by Falconer, that are closed under countable intersections, and such that every set in the class has Hausdorff dimension at least $s$. We provide a Frostman type lemma to determine if a limsup-set is in such a class. Suppose $E = \limsup E_n \subset [0,1]$, and that $μ_n$ are probability measures with support in $E_n$. If there is a constant $C$ such that \[\iint|x-y|^{-s}\, \mathrm{d}μ_n(x)\mathrm{d}μ_n(y)<C\] for all $n$, then under suitable conditions on the limit measure of the sequence $(μ_n)$, we pro…
We show that the sets of weighted badly approximable vectors in $\Bbb R^n$ are winning sets of certain games, which are modifications of $(α,β)$-games introduced by W. Schmidt in 1966. The latter winning property is stable with respect to countable intersections, and is shown to imply full Hausdorff dimension.
We present a comprehensive framework for the study of the size and large intersection properties of sets of limsup type that arise naturally in Diophantine approximation and multifractal analysis. This setting encompasses the classical ubiquity techniques, as well as the mass and the large intersection transference principles, thereby leading to a thorough description of the properties in terms of Hausdorff measures and large intersection classes associated with general gauge functions. The sets issued from eutaxic sequences of points and optimal regular systems may naturally be described with…
A dimensional mass transference principle from balls to open sets and applications to dynamical Diophantine approximation
本文提出了一种从球体到开集的维度质量转移原理，用于解决动力学迪奥菲尼逼近中的缩小目标问题，并揭示了热力学正式主义与该领域的新联系。
This text is a slightly expanded version of a survey article on certain aspects of low dimensional dynamics and number theory written after a kind invitation by the editors of the Notices of the American Mathematical Society.
Twisted Diophantine approximation on manifolds
本文研究了在流形上旋转变换的丢番图逼近问题，提出了非退化解析流形的充分条件，并探讨了badly approximable与well approximable向量与流形的交集性质。
We estalish the conjectures of Sprindzhuk over a local field of positive characteristic using the dynamical method of Kleinbock-Margulis.
For $λ\in (1/2, 1)$ and $α$, we consider sets of numbers $x$ such that for infinitely many $n$, $x$ is $2^{-αn}$-close to some $\sum_{i=1}^n ω_i λ^i$, where $ω_i \in \{0,1\}$. These sets are in Falconer's intersection classes for Hausdorff dimension $s$ for some $s$ such that $- \frac{1}α \frac{\log λ}{\log 2} \leq s \leq \frac{1}α$. We show that for almost all $λ\in (1/2, 2/3)$, the upper bound of $s$ is optimal, but for a countable infinity of values of $λ$ the lower bound is the best possible result.
On Diophantine equations involving intersection of Thabit and Williams numbers base $b$ and some ternary recurrent sequences
本文研究了 Padovan 数、Perrin 数和 Narayana's cows 数与基 $b$ 下 Thabit 数及 Williams 数相关的丢番图方程，确定了符合条件的数，并求出了 $2 \leq b \leq 10$ 范围内的所有解。
The nature and origin of exceptional sets associated with the rotation number of circle maps, Kolmogorov-Arnol'd-Moser theory on the existence of invariant tori and the linearisation of complex diffeomorphisms are explained. The metrical properties of these exceptional sets are closely related to fundamental results in the metrical theory of Diophantine approximation. The counterpart of Diophantine approximation in hyperbolic space and a dynamical interpretation which led to the very general notion of `shrinking targets' are sketched and the recent use of flows in homogeneous spaces of lattice…
It is shown that for any translation invariant outer measure M, the M-measure of the intersection of any subset of R^n that is invariant under rational translations and which does not have full Lebesgue measure with an the closure of an open set of positive measure cannot be positive and finite. Analogues for $p$-adic fields and fields of formal power series over a finite field are established. The results are applied to some problems in metric Diophantine approximation.
For two properly intersecting effective cycles in projective space X,Y, and their intersection product Z, the metric Bezout Theorem relates the degrees, heights of X,Y, and Z, as well as their distances and algebraic distances to a given point theta. Applications of this Theorem are in the area of Diophantine Approximation, giving estimates for approximation properties of Z with respect to $θ$ against the ones of X, and Y.
Metric results of the intersection of sets in Diophantine approximation
本文研究丢番图逼近中集合 $E(\psi)$ 的度量性质，计算其 $s$ 维 Hausdorff 测度，并证明多个此类集合乘积的 Hausdorff 维数公式。
Inhomogeneous Diophantine Approximation on $M_0$-sets
本文在 M_0-集上建立了带约束分母的定量 Schmidt 型逼近定理，通过分母增长与 Rajchman 测度傅里叶变换衰减之间的平衡条件，得到非齐次 Khintchine-Järnik 型结果及交集 Hausdorff 维数下界。
The metric Bezout Theorem proved in an earlier paper can be extended to a derivative version that compares derivatives of the algebraic distance of a point $θ$ to two properly intersecting cycles in projective space with the derivatives of the algebraic distance of $θ$ to their intersection. This improvement can be used to make algebraic independence criteria, to be proved in a forthcoming paper, more flexible and to refine Approximation results that are proved using the metric Bezout Theorem.
For any $i,j \ge 0$ with $i+j =1$, let $\bad(i,j)$ denote the set of points $(x,y) \in \R^2$ for which $ \max \{\|qx\|^{1/i}, \|qy\|^{1/j} \} > c/q $ for all $ q \in \N $. Here $c = c(x,y)$ is a positive constant. Our main result implies that any finite intersection of such sets has full dimension. This settles a conjecture of Wolfgang M. Schmidt in the theory of simultaneous Diophantine approximation.
Let $ϕ(z)$ be a non-isotrivial rational function in one-variable with coefficients in $\overline{\mathbb{F}}_p(t)$ and assume that $γ\in\mathbb{P}^1(\overline{\mathbb{F}}_p(t))$ is not a post-critical point for $ϕ$. Then we prove that the diophantine approximation exponent of elements of $ϕ^{-m}(γ)$ are eventually bounded above by $\lceil d^m/2\rceil+1$. To do this, we mix diophantine techniques in characteristic $p$ with the adelic equidistribution of small points in Berkovich space. As an application, we deduce a form of Silverman's celebrated limit theorem in this setting. Namely, if we tak…
Let $μ$ be a Gibbs measure of the doubling map $T$ of the circle. For a $μ$-generic point $x$ and a given sequence $\{r_n\} \subset \R^+$, consider the intervals $(T^nx - r_n \pmod 1, T^nx + r_n \pmod 1)$. In analogy to the classical Dvoretzky covering of the circle we study the covering properties of this sequence of intervals. This study is closely related to the local entropy function of the Gibbs measure and to hitting times for moving targets. A mass transference principle is obtained for Gibbs measures which are multifractal. Such a principle was shown by Beresnevich and Velani \cite{BV}…
Twisted Diophantine Approximation II: Uniform Theory
本文提出一种统一方法，利用对角格轨迹的逐次极小值精确计算扭曲一致丢番图逼近的Hausdorff维数，涵盖端点集、水平集及零满定律，并推广了多个已知结果。
Let $β>1$ be a real number and define the $β$-transformation on $[0,1]$ by $T_β:x\mapsto βx\bmod 1$. Further, define $$W_y(T_β,Ψ):=\{x\in [0, 1]:|T_β^nx-y|<Ψ(n) \mbox{ for infinitely many $n$}\}$$ and $$W(T_β,Ψ):=\{(x, y)\in [0, 1]^2:|T_β^nx-y|<Ψ(n) \mbox{ for infinitely many $n$}\},$$ where $Ψ:\mathbb{N}\to\mathbb{R}_{>0}$ is a positive function such that $Ψ(n)\to 0$ as $n\to \infty$. In this paper, we show that each of the above sets obeys a Jarník-type dichotomy, that is, the generalised Hausdorff measure is either zero or full depending upon the convergence or divergence of a certain serie…
Families of Unit Equations and Exponential Diophantine Problems via Integral Points
本文通过Ru-Vojta定理和Huang-Levin-Xiao不等式推广研究了投影簇上整数点的分布，证明了两类一参数单位方程解集的退化性，并扩展了GCD估计以获得特定指数迪菲antine方程和q进制表示中数字分布的新结果。
We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and Margulis ('98) resolving Sprindžuk's conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss ('04), hereafter abbreviated KLW. As applications we prove the extremality of all hyperbolic measures of smooth dynamical systems with sufficiently large Hausdorff dimension, and of the Patterson--Sullivan measures of all nonplanar geometrically finite…
Let $\mathcal{D}=(d_n)_{n=1}^\infty$ be a bounded sequence of integers with $d_n\ge 2$ and let $(i, j)$ be a pair of strictly positive numbers with $i+j=1$. We prove that the set of $x \in \RR$ for which there exists some constant $c(x) > 0$ such that \[ \max\{|q|_\DDD^{1/i}, \|qx\|^{1/j}\} > c(x)/ q \qquad \forall q \in \NN \] is one quarter winning (in the sense of Schmidt games). Thus the intersection of any countable number of such sets is of full dimension. In turn, this establishes the natural analogue of Schmidt's conjecture within the framework of the de Mathan-Teulié conjecture -- als…
We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove that the continued fraction expansion of almost any point on the middle third Cantor set (with respect to the natural measure) contains all finite patterns (hence is well approximable). Similarly, we show that for a variety of fractals in [0,1]^2, possessing some symmetry, almost any point is not Dirichlet improvable (hence is well approximable) and has property C (after Cassels). We then settle by similar methods a conjecture of M. Boshernitzan saying that there are no …
Let (X,d) be a metric space and (Ω, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of Ω. Loosely speaking, these consist of points in Ωwhich `stay clear' of some given set of points in X. The classical set \Bad of `badly approximable' numbers in the theory of Diophantine approximation falls within our framework as do the sets \Bad(i,j) of simultaneously badly approximable numbers. Under various natural conditions we prove that the badly approximable subsets of Ωhave full Hausdorff dimension. Applications of o…
Counting and Joint equidistribution of approximates
本文利用动力学技巧研究矩阵Diophantine逼近元的计数与联合等分布问题，通过在幺模格空间上构造Poincaré截面并引入处理高秩群作用的新方法，描述了极限测度，并首次处理了乘性逼近情形。
In 1995, Hill and Velani introduced the shrinking targets theory. Given a dynamical system $([0,1],T)$, they investigated the Hausdorff dimension of sets of points whose orbits are close to some fixed point. In this paper, we study the sets of points well-approximated by orbits $\{T^n x\}_{n\geq 0}$, where $T$ is an expanding Markov map with a finite partition supported by $[0,1]$. The dimensions of these sets are described using the multifractal properties of invariant Gibbs measures.
Circle coverings driven by arithmetic sequences: a percolation approach to Diophantine approximation and fractal intersections
研究通过缩小区间覆盖[0,1)的问题，揭示了在特定序列下覆盖半径的最优性，并扩展了对分形交集的尺寸计算。
We study almost prime solutions of systems of Diophantine equations in the Birch setting. Previous work shows that there exist integer solutions of size B with each component having no prime divisors below $B^{1/u}$, where $u=c_0n^{3/2}$, $n$ is the number of variables and $c_0$ is a constant depending on the degree and the number of equations. We improve the polynomial growth $$n^{3/2}$$ to the logarithmic $$\frac{\log n}{\log \log n}.$$ Our main new ingredients are the generalisation of the Brüdern-Fouvry vector sieve in any dimension and the incorporation of smooth weights into the Davenpor…
Bad is null
该研究构建了度量空间中坏逼近点的通用框架，证明温和假设下坏逼近点集测度为零，可应用于丢番图逼近、动力系统等多个相关领域。
Let $μ\geq 2$ be a real number and let $\Mcal(μ)$ denote the set of real numbers approximable at order at least $μ$ by rational numbers. More than eighty years ago, Jarní k and, independently, Besicovitch established that the Hausdorff dimension of $\Mcal(μ)$ is equal to $2/μ$. We investigate the size of the intersection of $\Mcal(μ)$ with Ahlfors regular compact subsets of the interval $[0, 1]$. In particular, we propose a conjecture for the exact value of the dimension of $\Mcal(μ)$ intersected with the middle-third Cantor set and give several results supporting this conjecture. We especiall…
Simultaneous Diophantine approximation on the three-dimensional Veronese curve: the complete Hausdorff dimension story
本研究补全了三维Veronese曲线与联立λ-良逼近点集交集的豪斯多夫维数的λ取值范围，给出λ≥1/3时的维数公式，使该曲线成为首个维数理论被完全确定的非退化非平面曲线。
The dimension of well approximable numbers
本文综述了受Besicovitch结果启发的丢番图逼近核心主题，重点介绍了Mass Transference Principle、Ubiquity及流形与分形上的逼近进展，并揭示了数论与分形几何的联系及开放问题。
The continuous version of a fundamental result of Khinchin says that a half-infinite torus line in the unit square $[0,1]^2$ exhibits superdensity, which is a best form of time-quantitative density, if and only if the slope of the geodesic is a badly approximable number. In this paper, we give a proof of the extension of this result of Khinchin to the case when the unit torus $[0,1]^2$ is replaced by a finite polysquare surface, or square tiled surface. The argument is based on diophantine approximation and continued fractions, traditional tools in number theory. In particular, we use the famo…
On some Diophantine and Ergodic properties of the Schneider map over function fields
该论文研究函数域上Schneider映射定义的连分数，证明其Laurent级数域元素的连分数展开存在唯一，确立相关丢番图逼近与遍历性质，确定最难逼近元素集的Hausdorff维数，为函数域数论与动力系统提供新结果。
Recent years have seen very important developments at the interface of Diophantine approximation and homogeneous dynamics. In the first part of the paper we give a brief exposition of a dictionary developed by Dani and Kleinbock-Margulis which relates Diophantine properties of vectors to distribution of orbits of flows on the space of unimodular lattices. In the second part of the paper we briefly describe an extension of this dictionary recently developed by the authors, which establishes an analogous dynamical correspondence for general lattice orbits on homogeneous spaces. We concentrate sp…
Khintchine's theorem and Diophantine approximation on manifolds
本文提出结合几何、动力学思想与“通有部分和特殊部分”新技术的新方法，证明了$\mathbb{R}^n$任意非退化子流形的收敛型欣钦型定理，细化得到相关豪斯多夫维数精确值，解决了丢番图逼近领域的长期问题。
We introduce a metric of mutual energy for adelic measures associated to the Arakelov-Zhang pairing. Using this metric and potential theoretic techniques involving discrete approximations to energy integrals, we prove an effective bound on a problem of Baker and DeMarco on unlikely intersections of dynamical systems, specifically, for the set of complex parameters $c$ for which $z=0$ and $1$ are both preperiodic under iteration of $f_c(z)=z^2 + c$.
We show that for any $ε<1$ and any $\mathcal{T}$ `drifting away from walls', Dirichlet's Theorem cannot be $ε$-improved along $\mathcal{T}$ for Lebesgue almost every system of linear forms $Y$ (see the paper for definitions). In the case $m = 1$ we also show that for a large class of measures $μ$ there is $ε_0>0$ such that for any drifting away from walls $\mathcal{T}$, any $ε<ε_0$, and for $μ$-almost every $Y$, Dirichlet's Theorem cannot be $ε$-improved along $\mathcal{T}$. These measures include natural measures on sufficiently regular smooth manifolds and fractals. Our results extend those …
We prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection, then the polynomials have a common iterate. We also prove a special case of a conjectured dynamical analogue of the Mordell-Lang conjecture.
Diophantine approximation and the Mass Transference Principle: incorporating the unbounded setup
本文将 Wang 与 Wu 的矩形质量转移原理推广到边长低阶为无穷大的无界情形，并据此确定联立丢番图逼近和收缩目标问题中自然 limsup 集的 Hausdorff 维数。
In this notes we make a comparison between the arithmetic properties of irrational numbers and their dynamical properties under the Gauss map. We show some equivalences between different classifications of irrational numbers such as the Diophantine classes and numbers admitting approximations by rational numbers at a given 'speed'. We also show that irrational numbers with finite upper Lyapunov exponent for the Gauss map satisfy a Diophantine condition.
The dynamical Mordell-Lang conjecture concerns the structure of the intersection of an orbit in an algebraic dynamical system and an algebraic variety. In this paper, we bound the size of this intersection for various cases when it is finite.
This paper, motivated by problems in Diophantine analysis which can be formulated as problems of finding rational points on the intersection of two quadrics, presents an explicit construction of a rationally defined isomorphism (biregular mapping) between a rationally defined smooth intersection of two quadrics in projective three-space and an elliptic curve in Weierstrass form which maps a distinguished rational point to the point at infinity. The usual approach of transforming a smooth plane cubic to a curve in Weierstrass form by mapping an inflection point to the point at infinity in a par…
Badly approximable points on non-linear carpets
本文研究了非线性非共形吸引子中难近点的全维相交性质，并给出了这些吸引子的Hausdorff维数公式。
We study a norm sensitive Diophantine approximation problem arising from the work of Davenport and Schmidt on the improvement of Dirichlet's theorem. Its supremum norm case was recently considered by the first-named author and Wadleigh, and here we extend the set-up by replacing the supremum norm with an arbitrary norm. This gives rise to a class of shrinking target problems for one-parameter diagonal flows on the space of lattices, with the targets being neighborhoods of the critical locus of a suitably scaled norm ball. We use methods from geometry of numbers and dynamics to generalize a res…
Rigidity and unlikely intersections for stable $p$-adic dynamical systems
针对 Berger 关于稳定 $p$-进幂级数预周期点能否确定稳定 $p$-进动力系统的问题，本文利用这些预周期点实现了对相应动力系统的确定。
Diophantine approximations for translation surfaces and planar resonant sets
本文利用平移曲面产生的平面共振集，研究 Teichmüller 测地线有界性、规定速率偏移及有理台球快速回复方向的 Hausdorff 维数，建立了良好逼近与不良逼近方向的二分法及维数估计。
The Lonely Runner Conjecture is a number theory problem, dating to 1964. Using dynamical systems theory, we show almost all sets of velocities solve the conjecture. Furthermore, any "traditional" approach of Diophantine approximation cannot solve the problem, and we offer a short list of reformulations of the problem.
Solutions of the quartic Fermat equation in ring class fields of odd conductor over quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ with $-d \equiv 1$ (mod $8$) are shown to be periodic points of a fixed algebraic function $T(z)$ defined on the punctured disk $0< |z|_2 \le \frac{1}{2}$ of the maximal unramified, algebraic extension $\textsf{K}_2$ of the $2$-adic field $\mathbb{Q}_2$. All ring class fields of odd conductor over imaginary quadratic fields in which the prime $p=2$ splits are shown to be generated by complex periodic points of the algebraic function $T$, and conversely, all but two of …
We study families of varieties endowed with polarized canonical eigensystems of several maps, inducing canonical heights on the dominating variety as well as on the "good" fibers of the family. We show explicitely the dependence on the parameter for global and local canonical heights defined by Kawaguchi when the fibers change, extending previous works of J. Silverman and others. Finally, fixing an absolute value $v \in K$ and a variety $V/K$, we descript the Kawaguchi`s canonical local height $\hatλ_{V,E,\mathcal{Q},}(.,v)$ as an intersection number, provided that the polarized system $(V,\ma…
Density of the multidimensional Lagrange spectrum
本文证明多维拉格朗日谱的闭包为0与其上确界的区间，利用丢番图逼近与幺模格空间动力学的对应关系，结合horosphere膨胀平移的均匀分布完成证明。
We prove that the countable intersection of $C^1$-diffeomorphic images of certain Diophantine sets has full Hausdorff dimension. For example, we show this for the set of badly approximable vectors in $\mathbb{R}^d$, improving earlier results of Schmidt and Dani. To prove this, inspired by ideas of McMullen, we define a new variant of Schmidt's $(α, β)$-game and show that our sets are hyperplane absolute winning (HAW), which in particular implies winning in the original game. The HAW property passes automatically to games played on certain fractals, thus our sets intersect a large class of frac…
Lagrange spectrum for Diophantine approximations of complex numbers with real part equal to one half
该研究在A. Schmidt相关定理启发下，刻画实部为1/2的复数的受限拉格朗日谱，推导其分形性质并证明满足特定逼近下界的复数集不可数。
We prove an analogue the Khinchin theorem for the Diophantine approximation by integer vectors lying on a quadratic variety. The proof is based on the study of a dynamical system on a homogeneous space of the orthogonal group. We show that in this system, generic trajectories visit a family of shrinking subsets infinitely often.
Let f:X->X be a morphism of a variety over a number field K. We consider local conditions and a "Bruaer-Manin" condition, defined by Hsia and Silverman, for the orbit of a point P in X(K) to be disjoint from a subvariety V of X, i.e., the intersection of the orbit of P with V is empty. We provide evidence that the dynamical Brauer-Manin condition is sufficient to explain the lack of points in the intersection of the orbit of P with V; this evidence stems from a probabilistic argument as well as unconditional results in the case of etale maps.
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