Mumtaz Hussain, Simon Kristensen
In this paper we show that the set of mixed type badly approximable simultaneously small linear forms is of maximal dimension. As a consequence of this theorem we settle a conjecture of the first author.
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Given an $a$-dimensional linear subspace $\mathfrak{A}$ in $\mathbb{R}^d$ which contains a badly approximable $b$-dimensional subspace $\mathfrak{B} \subset \mathfrak{A}$. We study the badly approximability almost all $c$-dimensional linear subspaces $\mathfrak{C} \subset \mathfrak{A}$.
We prove that the set of (r_1,r_2,..,r_{d})-badly approximable vectors is a winning set if r_1=r_2=...=r_{d-1}\geq r_{d}.
Badly approximable systems of linear forms over a field of formal series
证明了系数来自有限域的洛朗级数域上的m个n元线性型的差逼近系统的豪斯多夫维数是极大的.这是一个类似施密特的多维概括Jarnik定理的严重逼近的数字。
Some explicit badly approximable pairs
考虑了一对无理数的具有公分母的超范数同时有理逼近的丢番图逼近问题，并对某些具有大逼近常数的无理数对进行了显式计算.其中一对是迄今为止计算出的最差的近似对。
We provide a number of conditions on the rational numbers $u$ and $v$ which ensure that the Laurent series $g_{u,v}(x):=\prod_{t=0}^\infty (1+ux^{-3^t} + vx^{-2\cdot 3^t})$ is badly approximable.
We study the notion of strongly badly approximable matrices in the field of power series over a field $K$. We prove a transference principle in this setting, and show that such matrices exist when $K$ is infinite.
We give a sketch for an alternative proof of a recent result by J. Tseng.
We call a badly approximable number $decaying$ if, roughly, the Lagrange constants of integer multiples of that number decay as fast as possible. In this terminology, a question of Y. Bugeaud ('15) asks to find the Hausdorff dimension of the set of decaying badly approximable numbers, and also of the set of badly approximable numbers which are not decaying. We answer both questions, showing that the Hausdorff dimensions of both sets are equal to one. Part of our proof utilizes a game which combines the Banach--Mazur game and Schmidt's game, first introduced in Fishman, Reams, and Simmons (prep…
We establish a strong form of Littlewood's conjecture with inhomogeneous shifts, for a full-dimensional set of pairs of badly approximable numbers on a vertical line. We also prove a uniform assertion of this nature, generalising a strong form of a result by Haynes, Jensen and Kristensen. Finally, we establish a similar result involving inhomogeneously badly approximable numbers, making progress towards a problem posed by Pollington, Velani, Zafeiropoulos and Zorin.
We explore and refine techniques for estimating the Hausdorff dimension of exceptional sets and their diffeomorphic images. Specifically, we use a variant of Schmidt's game to deduce the strong C^1 incompressibility of the set of badly approximable systems of linear forms as well as of the set of vectors which are badly approximable with respect to a fixed system of linear forms.
We consider approximation of vectors $\mathbf{z}\in F\otimes\mathbb{R}\cong\mathbb{R}^r\times\mathbb{C}^s$ by elements of a number field $F$ and construct examples of badly approximable vectors. These examples come from compact subspaces of $SL_2(\mathcal{O}_F)\backslash SL_2(F\otimes\mathbb{R})$ naturally associated to (totally indefinite, anisotropic) $F$-rational binary quadratic and Hermitian forms, a generalization of the well-known fact that quadratic irrationals are badly approximable over $\mathbb{Q}$.
This paper is motivated by Davenport's problem and the subsequent work regarding badly approximable points in submanifolds of a Euclidian space. We study the problem in the area of twisted Diophantine approximation and present two different approaches. The first approach shows that, under a certain restriction, any countable intersection of the sets of weighted badly approximable points on any non-degenerate C^1 submanifold of R^n has full dimension. In the second approach we introduce the property of isotropically winning and show that the sets of weighted badly approximable points are isotro…
We prove that for any pair $(s,t)$ of nonnegative numbers with $s+t=1$, the set of two-dimensional $(s,t)$-badly approximable vectors is winning for Schmidt's game. As a consequence, we give a direct proof of Schmidt's conjecture using his game.
Badly approximable numbers related to the Littlewood conjecture
利用Peres-Schlag方法证明了实数$α, β$的存在性，使得$$ \liminf_{q\to \infty}（q \log^2 q）||αq||||βq||> 0。
Motivated by a wonderful paper by Ngoc Ai Van Nguyen, Anthony Poëls and Damien Roy, where a powerful method was introduced, we prove a criterion for a vector $\pmbα\in \mathbb{R}^d$ to be a badly approximable vector. Moreover we construct certain examples which show that a more general version of our criterion is not valid.
We prove that Thue-Morse constant $τ_{TM}=0.01101001..._2$ is not a badly approximable number. Moreover, we prove that $τ_{TM}(a)=0.01101001..._a$ is not badly approximable for every integer base $a\geq 2$ such that $a$ is not divisible by 15. At the same time we provide a precise formula for convergents of the Laurent series $\tilde{f}_{TM}(z) = z^{-1}\prod_{n=1}^\infty (1-z^{-2^n})$, thus developing further the research initiated by Alf van der Poorten and others.
This paper is motivated by two problems in the theory of Diophantine approximation, namely, Davenport's problem regarding badly approximable points on submanifolds of a Euclidean space and Schmidt's problem regarding the intersections of the sets of weighted badly approximable points. The problems have been recently settled in dimension two but remain open in higher dimensions. In this paper we develop new techniques that allow us to tackle them in full generality. The techniques rest on lattice points counting and a powerful quantitative result of Bernik, Kleinbock and Margulis. The main theo…
Let $K$ be a number field, let $S$ be the set of all normalized, non-conjugate Archimedean valuations of $K$, and let $K_{S} = \prod_{v \in S} K_v$ be the Minkowski space associated with $K$. We strengthen recent results of \cite{EsdahlKristensen10} and \cite{EinsiedlerGhoshLytle13} by showing that the set of badly approximable elements of $K_S$ is $\mathcal{H}$-absolute winning for a certain family of subspaces of $K_{S}$.
Badly approximable affine forms and Schmidt games
对任何实数\t，所有实数x的集合，其中存在常数c（x）> 0，使得\inf_{p \in \ZZ}| \t q - x - p|对于\ZZ {0}中的所有q，\geq c（x）/ |q|是1/8获胜集。
Delone sets associated with badly approximable triangles
本文构造与难逼近数相关、预期具旋转不变衍射的新 Delone 集，并借助方程 $x+y+z=1$ 优化瓦片取向差异，证明恰有两个字典序下部分商最小的解。
For any pair of real numbers $(i,j)$ with $0<i,j<1$ and $i+j=1$, we prove that the set of $p$-adic mixed $(i,j)$-badly approximable numbers $\bad_p(i, j)$ is 1/2-winning in the sense of Schmidt's game. This improves a recent result of Badziahin, Levesley, and Velani on mixed Schmidt conjecture.
Consider irrational affine subspace $ A\subset \mathbb{R}^d$ of dimension $a$. We prove that the set $$ \{ξ=(ξ_1,...,ξ_d) \in {A}:\,\,\, \ q^{1/a}\cdot \max_{1\le i \le d} ||qξ_i|| \to \infty,\,\,\,\, q\to \infty\} $$ is an $α$-winning set for every $α\in (0,1/2]$
We prove a result in the area of twisted Diophantine approximation related to the theory of Schmidt games. In particular, under certain restrictions we give a affirmative answer to the analogue in this setting of a famous conjecture of Schmidt from Diophantine approximation.
The Littlewood Conjecture states that liminf_{q\to \infty} q . ||qx|| . ||qy|| = 0 for all pairs (x,y) of real numbers. We show that with the additional factor of log q . loglog q the statement is false. Indeed, our main result implies that the set of (x,y) for which liminf_{q\to\infty} q . log q . loglog q . ||qx|| . ||qy|| > 0 is of full dimension.
In this paper, we study the weighted $n$-dimensional badly approximable points on manifolds. Given a $C^n$ differentiable non-degenerate submanifold $\mathcal{U} \subset \mathbb{R}^n$, we will show that any countable intersection of the sets of the weighted badly approximable points on $\mathcal{U}$ has full Hausdorff dimension. This strengthens a result of Beresnevich by removing the condition on weights and weakening the smoothness condition on manifolds. Compared to the work of Beresnevich, our approach relies on homogeneous dynamics. It turns out that in order to solve this problem, it is …
In this paper we investigate the problem of how well points in finite dimensional p-adic solenoids can be approximated by rationals. The setting we work in was previously studied by Palmer, who proved analogues of Dirichlet's theorem and the Duffin-Schaeffer theorem. We prove a complementary result, showing that the set of badly approximable points has maximum Hausdorff dimension. Our proof is a simple application of the elegant machinery of Schmidt's game.
We establish that the set of pairs $(α, β)$ of real numbers such that $$ \liminf_{q \to + \infty} q \cdot (\log q)^2 \cdot \Vert q α\Vert \cdot \Vert q β\Vert > 0, $$ where $\Vert \cdot \Vert$ denotes the distance to the nearest integer, has full Hausdorff dimension in $\R^2$. Our proof rests on a method introduced by Peres and Schlag, that we further apply to various Littlewood-type problems
Schmidt's game, Badly Approximable Linear Forms and Fractals
证明了对任意两个自然数M和N，若Tau是R^MN的紧集K上的Borel有限绝对友好测度，则K与BA（M，N）的交集是K上Schmidt博弈意义下的获胜集，其中BA（M，N）是M \times N矩阵的差逼近集.作为直接结果，我们有以下应用。如果K是满足开集条件的R^（M \times N）的不可约有限族压缩相似映射的吸引子（例如Cantor三元集、Koch曲线和Sierpinski垫片），则dimK=dimK \capBA（M，N）。
We study the generalized continued fraction expansions of complex numbers in term of elements from Euclidean subrings, especially Gaussian or Eisenstein integers, in a general framework as pursued in [3] and [1]. We introduce a common approach to studying the continued fraction expansions of zeroes of binary forms, via consideration of the action of the general linear group, and apply it to discuss expansions of quadratic surds on the one hand and of badly approximable numbers on the other hand. Also, we generalize the property of the simple continued fraction expansions that the convergents a…
We give an optimal version of the classical ``three-gap theorem'' on the fractional parts of $n θ$, in the case where $θ$ is an irrational number that is badly approximable. As a consequence, we deduce a version of Kronecker's inhomogeneous approximation theorem in one dimension for badly approximable numbers. We apply these results to obtain an improved measure of sequence diversity for characteristic Sturmian sequences, where the slope is badly approximable.
Given a badly approximable number $α$, we study the asymptotic behaviour of the Sudler product defined by $P_N(α) = \prod_{r=1}^N 2 | \sin πr α|$. We show that $\liminf_{N \to \infty} P_N(α) = 0$ and $\limsup_{N \to \infty} P_N(α)/N = \infty$ whenever the sequence of partial quotients in the continued fraction expansion of $α$ exceeds $7$ infinitely often. This improves results obtained by Lubinsky for the general case, and by Grepstad, Neumüller and Zafeiropoulos for the special case of quadratic irrationals. Furthermore, we prove that this threshold value $7$ is optimal, even when restrictin…
A badly approximable system of affine forms is determined by a matrix and a vector. We show Kleinbock's conjecture for badly approximable systems of affine forms: for any fixed vector, the set of badly approximable systems of affine forms is winning (in the sense of Schmidt games) even when restricted to a fractal (from a certain large class of fractals). In addition, we consider fixing the matrix instead of the vector where an analog statement holds.
Let p be a prime number. The p-adic case of the Mixed Littlewood Conjecture states that liminf_{q \to \infty} q . |q|_p . ||q x|| = 0 for all real numbers x. We show that with the additional factor of log q.loglog q the statement is false. Indeed, our main result implies that the set of x for which liminf_{q\to\infty} q . log q . loglog q. |q|_p . ||qx|| > 0 is of full dimension. The result is obtained as an application of a general framework for Cantor sets developed in this paper.
The dimension of the set of $ψ$-badly approximable points in all ambient dimensions; on a question of Beresnevich and Velani
该研究解决Beresnevich与Velani的相关问题，证明任意环境维度下$\psi$-坏逼近点集的Hausdorff维数与$\psi$-良逼近点集一致，采用基于延迟修剪的质量转移原理新证法，推广了一维已有结果。
Diophantine approximation and badly approximable sets
设（X，d）是一个度量空间，（Ω，d）是X的一个紧致子空间，它支持一个非原子有限测度m.我们认为“自然”类的Ω的严重逼近的子集。不严格地说，它们由Ω中的点组成，这些点与X中的某个给定点集“保持清晰”。经典的集合\Bad的“严重逼近”的数字在理论的丢番图逼近属于我们的框架做集合\Bad（i，j）的同时严重逼近的数字。在各种自然条件下，我们证明了Ω的差逼近子集具有满豪斯多夫维数.我们的一般框架的应用包括那些从数论（古典，复杂，p-adic和正式的幂级数）和动力系统（迭代函数计划，合理的映射和Kleinian群）。
Hausdorff dimension of differences of badly approximable sets
针对不可良好逼近集差集的维数问题，通过在幺模网格空间上使用施密特博弈的变体“快速博弈”，证明该集差具有满豪斯多夫维数，补充了相关领域的维数研究结果。
Let C be two times continuously differentiable curve in R^2 with at least one point at which the curvature is non-zero. For any i,j > 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 integers q >0. Here c = c(x,y) is a positive constant. Our main result implies that any finite intersection of such sets with C has full Hausdorff dimension. This provides a solution to a problem of Davenport dating back to the sixties.
The set of badly approximable $m \times n $ matrices is known to have Hausdorff dimension $mn $. Each such matrix comes with its own approximation constant $c$, and one can ask for the dimension of the set of badly approximable matrices with approximation constant greater than or equal to some fixed $c$. In the one-dimensional case, a very precise answer to this question is known. In this note, we obtain upper and lower bounds in higher dimensions.
Cycle integrals of modular functions are expected to play a role in real quadratic analogue of singular moduli. In this paper, we extend the definition of cycle integrals of modular functions from real quadratic numbers to badly approximable numbers. We also give explicit representations of values of extended-cycle integrals for some cases.
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.
Badly approximable systems of affine forms
我们证明了W的一个非齐次模拟。M. Schmidt（1969）关于线性型的坏逼近系统的集合的豪斯多夫维数的定理。证明是基于动力系统理论的思想和方法，特别是基于李群的齐性空间上混合流的有界轨道的丰富性。
For a real number $x$, $\| x\| = \min \{|x-p|: p\in Z\}$ is the distance of $x$ to the nearest integer. We say that two real numbers $θ$, $θ'$ are $\pm$ equivalent if their sum or difference is an integer. Let $θ$ be irrational and put \[ϕ(θ) = \inf \{q \,\| q θ\| : q \in N \}. \] We will prove: If $ϕ(θ)> 1/3$, then $θ$ is $\pm$ equivalent to a root of $f_m (x,1) = 0$, where $f_m$ is a Markoff form. Conversely, if $θ$ is $\pm$ equivalent to a root of $f_m(x,1)=0$, then \[ϕ(θ) = m \| mθ\| = \frac{2}{3+\sqrt{9-4m^{-2}}} > 1/3. \]
In this paper we prove that badly approximable points on any analytic non-degenerate curve in $\mathbb{R}^n$ is an absolute winning set. This confirms a key conjecture in the area stated by Badziahin and Velani (2014) which represents a far-reaching generalisation of Davenport's problem from the 1960s. Amongst various consequences of our main result is a solution to Bugeaud's problem on real numbers badly approximable by algebraic numbers of arbitrary degree. The proof relies on new ideas from fractal geometry and homogeneous dynamics.
For any real pair i, j geq 0 with i+j=1 let Bad(i, j) denote the set of (i, j)-badly approximable pairs. That is, Bad(i, j) consists of irrational vectors x:=(x_1, x_2) in R^2 for which there exists a positive constant c(x) such that max {||qx_1||^(-i), ||qx_2||^(-j)} > c(x)/q for all q in N. A new characterization of the set Bad(i, j) in terms of `well-approximable' vectors in the area of 'twisted' inhomogeneous Diophantine approximation is established. In addition, it is shown that Bad^x(i, j), the `twisted' inhomogeneous analogue of Bad(i, j), has full Hausdorff dimension 2 when x is chosen…
We recall the notion of nearest integer continued fractions over the Euclidean imaginary quadratic fields $K$ and characterize the "badly approximable" numbers, ($z$ such that there is a $C(z)>0$ with $|z-p/q|\geq C/|q|^2$ for all $p/q\in K$), by boundedness of the partial quotients in the continued fraction expansion of $z$. Applying this algorithm to "tagged" indefinite integral binary Hermitian forms demonstrates the existence of entire circles in $\mathbb{C}$ whose points are badly approximable over $K$, with effective constants. By other methods (the Dani correspondence), we prove the exi…
We prove that for all $b$, the Hausdorff dimension of the set of $m \times n$ matrices $ε$-badly approximable for the target $b$ is not full. The doubly metric case follows. It was known that for almost every matrix $A$, the Hausdorff dimension of the set $Bad_A(ε)$ of $ε$-badly approximable target $b$ is not full, and that for real numbers $α$, $\dim_H Bad_α(ε)=1$ if and only if $α$ is singular on average. We show that if $\dim_H Bad_A(ε)=m$, then $A$ is singular on average.
For any j_1,...,j_n>0 with j_1+...+j_n=1 and any x \in R^n, we consider the set of points y \in R^n for which max_{1\leq i\leq n}(||qx_i-y_i||^{1/j_i})>c/q for some positive constant c=c(y) and all q\in N. These sets are the `twisted' inhomogeneous analogue of Bad(j_1,...,j_n) in the theory of simultaneous Diophantine approximation. It has been shown that they have full Hausdorff dimension in the non-weighted setting, i.e provided that j_i=1/n, and in the weighted setting when x is chosen from Bad(j_1,...,j_n). We generalise these results proving the full Hausdorff dimension in the weighted se…
For given $ε>0$ and $b\in\mathbb{R}^m$, we say that a real $m\times n$ matrix $A$ is $ε$-badly approximable for the target $b$ if $$\liminf_{q\in\mathbb{Z}^n, \|q\|\to\infty} \|q\|^n \langle Aq-b \rangle^m \geq ε,$$ where $\langle \cdot \rangle$ denotes the distance from the nearest integral point. In this article, we obtain upper bounds for the Hausdorff dimensions of the set of $ε$-badly approximable matrices for fixed target $b$ and the set of $ε$-badly approximable targets for fixed matrix $A$. Moreover, we give an equivalent Diophantine condition of $A$ for which the set of $ε$-badly appr…
Continued fraction expansions of complex numbers, Lagrange's theorem, and badly approximable numbers
本文通过推广连分数展开概念，研究复数的连分数展开性质，并扩展了拉格朗日定理及Hines关于极差近似复数的结果。
Random walks with badly approximable numbers
使用差异度量，我们分析了随机游动在d旋转产生的圆上的收敛速率，并建立了尖锐的速度，表明R^d中的d-元组的可近似性差会产生最快收敛的游动。我们使用差异度量，因为行走不会在全变差中收敛。对于可近似性差的d-元组，差异的上下界为（常数）k^（-d/2），其中k是随机游动的步数。我们展示了常数如何依赖于d元组。
For any i,j>0 with i+j =1, let Bad(i,j) denote the set of points (x,y) \in R^2 such that max \{ ||qx||^{1/i}, \, ||qy||^{1/j} \} > c/q for some positive constant c = c(x,y) and all q in N. We show that \Bad(i,j) \cap C is winning in the sense of Schmidt games for a large class of planar curves C, namely, everywhere non-degenerate planar curves and straight lines satisfying a natural Diophantine condition. This strengthens recent results solving a problem of Davenport from the sixties. In short, within the context of Davenport's problem, the winning statement is best possible. Furthermore, we o…
Approximation in this paper is of vectors on the unit $d$-cube by the projection of integer lattice points onto the same cube. We define badly approximable vectors on a rational quadratic variety and show that sets of these vectors, which are (naturally) indexed by $m \in \QQ$, are winning and strong winning in the sense of Schmidt games. From the winning property, it follows that these sets have full Hausdorff dimension and, moreover, so does their intersection. In most cases, these sets are known to be null sets.
On The Hausdorff Dimension of Weighted Badly Approximable Vectors
本文研究加权不良近似向量的豪斯多夫维数，通过扩展Koivusalo等人在无权情况下的构造和质量分布论证，证明了加权情况下的维数与无权情况相同。
Infinitely badly approximable affine forms
本文针对实矩阵与向量构成的仿射形式，提出新的奇异性概念来刻画无限坏逼近性质，并计算了相应集合的 Hausdorff 维数，同时给出其在网格空间中的动力系统解释。
For a proper, geodesic, Gromov hyperbolic metric space X, a discrete subgroup of isometries Γwhose limit set is uniformly perfect, and a disjoint collection of horoballs {H_j}, we show that the set of limit points badly approximable by {H_j} is absolutely winning in the limit set. As an application, we deduce that for a geometrically finite Kleinian group acting on H^{n+1}, the limit points badly approximable by parabolics is absolutely winning, generalizing previous results of Dani and McMullen. As a consequence of winning, we show that the set of badly approximable limit points has dimension…
Weighted badly approximable complex vectors and bounded orbits of certain diagonalizable flows
本文将加权坏逼近向量研究推广到全虚数域，证明SL2(C)乘积空间中在单参数Ad-半单子群作用下有界轨道的点构成超平面绝对获胜集，并推广了坏逼近复数集合的相关结果。
It is a classical result from Diophantine approximation that the set of badly approximable numbers has Lebesgue measure zero. In this paper we generalise this result to more general sequences of balls. Given a countable set of closed $d$-dimensional Euclidean balls $\{B(x_{i},r_{i})\}_{i=1}^{\infty},$ we say that $α\in \mathbb{R}^{d}$ is a badly approximable number with respect to $\{B(x_{i},r_{i})\}_{i=1}^{\infty}$ if there exists $κ(α)>0$ and $N(α)\in\mathbb{N}$ such that $α\notin B(x_{i},κ(α)r_{i})$ for all $i\geq N(α)$. Under natural conditions on the set of balls, we prove that the set of…
In this paper we prove inequalities for multiplicative analogues of Diophantine exponents, similar to the ones known in the classical case. Particularly, we show that a matrix is badly approximable if and only if its transpose is badly approximable and establish some inequalities connecting multiplicative exponents with ordinary ones.
We prove that if $J$ is the limit set of an irreducible conformal iterated function system (with either finite or countably infinite alphabet), then the badly approximable vectors form a set of full Hausdorff dimension in $J$. The same is true if $J$ is the radial Julia set of an irreducible meromorphic function (either rational or transcendental). The method of proof is to find subsets of $J$ that support absolutely friendly and Ahlfors regular measures of large dimension. In the appendix to this paper, we answer a question of Broderick, Kleinbock, Reich, Weiss, and the second-named author ('…
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。
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