Metric Diophantine approximation with respect to planar distance functions

平面直线上的乘法丢番图逼近与限制分母

Multiplicative Diophantine Approximation on Planar Lines with Restricted Denominators

本文研究了平面直线上具有限制分母的乘法丢番图逼近问题，推导了收敛条件并给出了Hausdorff维度的上界。

Diophantine approximation, Khintchine's theorem, torus geometry and Hausdorff dimension

A general form of the Borel-Cantelli Lemma and its connection with the proof of Khintchine's Theorem on Diophantine approximation and the more general Khintchine-Groshev theorem are discussed. The torus geometry in the planar case allows a relatively direct proof of the planar Groshev theorem for the set of $ψ$-approximable points in the plane. The construction and use of Haudsorff measure and dimension are explained and the notion of ubiquity, which is effective in estimating the lower bound of the Hausdorff dimension for quite general lim sup sets, is described. An application is made to obt…

Non-planarity and metric Diophantine approximation for systems of linear forms

In this paper we develop a general theory of metric Diophantine approximation for systems of linear forms. A new notion of `weak non-planarity' of manifolds and more generally measures on the space of $m\times n$ matrices over $\Bbb R$ is introduced and studied. This notion generalises the one of non-planarity in $\Bbb R^n$ and is used to establish strong (Diophantine) extremality of manifolds and measures. The notion of weak non-planarity is shown to be `near optimal' in a certain sense. Beyond the above main theme of the paper, we also develop a corresponding theory of inhomogeneous and weig…

Explicit bounds for rational points near planar curves and metric Diophantine approximation

The primary goal of this paper is to complete the theory of metric Diophantine approximation initially developed in [Ann. of Math.(2) 166 (2007), p.367-426] for $C^3$ non-degenerate planar curves. With this goal in mind, here for the first time we obtain fully explicit bounds for the number of rational points near planar curves. Further, introducing a perturbational approach we bring the smoothness condition imposed on the curves down to $C^1$ (lowest possible). This way we broaden the notion of non-degeneracy in a natural direction and introduce a new topologically complete class of planar cu…

A note on three problems in metric Diophantine approximation

The use of Hausdorff measures and dimension in the theory of Diophantine approximation dates back to the 1920s with the theorems of Jarnik and Besicovitch regarding well-approximable and badly-approximable points. In this paper we consider three inhomogeneous problems that further develop these classical results. Firstly, we obtain a Jarnik type theorem for the set of multiplicatively approximable points in the plane. This Hausdorff measure statement does not reduce to Gallagher's Lebesgue measure statement as one might expect and is new even in the homogeneous setting. Next, we establish a Ja…

A Jarník type theorem for planar curves: everything about the parabola

The well known theorems of Khintchine and Jarník in metric Diophantine approximation provide comprehensive description of the measure theoretic properties of real numbers approximable by rational numbers with a given error. Various generalisations of these fundamental results have been obtained for other settings, in particular, for curves and more generally manifolds. In this paper we develop the theory for planar curves by completing the theory in the case of parabola. This represents the first comprehensive study of its kind in the theory of Diophantine approximation on manifolds.

Inhomogeneous Diophantine approximation on curves and Hausdorff dimension

The goal of this paper is to develop a coherent theory for inhomogeneous Diophantine approximation on curves in $R^n$ akin to the well established homogeneous theory. More specifically, the measure theoretic results obtained generalize the fundamental homogeneous theorems of R.C. Baker (1978), Dodson, Dickinson (2000) and Beresnevich, Bernik, Kleinbock, Margulis (2002). In the case of planar curves, the complete Hausdorff dimension theory is developed

An Inhomogeneous Jarník type theorem for planar curves

In metric Diophantine approximation there are two main types of approximations: simultaneous and dual for both homogeneous and inhomogeneous settings. The well known measure-theoretic theorems of Khintchine and Jarník are fundamental in these settings. Recently, there has been substantial progress towards establishing a metric theory of Diophantine approximations on manifolds. In particular, both the Khintchine and Jarník type results have been established for planar curves except for only one case. In this paper, we prove an inhomogeneous Jarník type theorem for convergence on planar curves a…

A note on simultaneous Diophantine approximation on planar curves

Let $\cS_n(ψ_1,...,ψ_n)$ denote the set of simultaneously $(ψ_1,...,ψ_n)$--approximable points in $\R^n$ and $\cSM_n(ψ)$ denote the set of multiplicatively $ψ$--approximable points in $\R^n$. Let $\cM$ be a manifold in $\R^n$. The aim is to develop a metric theory for the sets $ \cM \cap \cS_n(ψ_1,...,ψ_n) $ and $ \cM \cap \cSM_n(ψ) $ analogous to the classical theory in which $\cM$ is simply $\R^n$. In this note, we mainly restrict our attention to the case that $\cM$ is a planar curve $\cC$. A complete Hausdorff dimension theory is established for the sets $ \cC \cap \cS_2(ψ_1,ψ_2) $ and $ \…

平移曲面与平面共振集的丢番图逼近

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。

Diophantine approximation on curves

Let $g$ be a dimension function. The Generalised Baker-Schmidt Problem (1970) concerns the $g$-dimensional Hausdorff measure ($\HH^g$-measure) of the set of $Ψ$-approximable points on nondegenerate manifolds. The problem relates the `size' of the set of $Ψ$-approximable points with the convergence or divergence of a certain series. In the dual approximation setting, the divergence case has been established by Beresnevich-Dickinson-Velani (2006) for any nondegenerate manifold. The convergence case, however, represents a major challenging open problem and progress thus far has been effectuated i…

Rational points near manifolds and metric Diophantine approximation

This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarnik type theorems for submanifolds of R^n. These problems have attracted a lot of interest since Kleinbock and Margulis proved a related conjecture of Alan Baker and V.G. Sprindzuk. They have been settled for planar curves but remain open in higher dimensions. In this paper, Khintchine and Jarnik type divergence theorems are established for arbitrary analytic non-degenerate manifolds regardless of their dimension. The key to establishing these results is the study o…

Simultaneous two-dimensional best Diophantine approximations in the Euclidean norm

We prove a new lower bound for the exponent of growth of the best two-dimensional Diophantine approximations with respect to Euclidean norm.

Diophantine properties of measures invariant with respect to the Gauss map

Motivated by the work of D. Y. Kleinbock, E. Lindenstrauss, G. A. Margulis, and B. Weiss, we explore the Diophantine properties of probability measures invariant under the Gauss map. Specifically, we prove that every such measure which has finite Lyapunov exponent is extremal, i.e. gives zero measure to the set of very well approximable numbers. We show on the other hand that there exist examples where the Lyapunov exponent is infinite and the invariant measure is not extremal. Finally, we answer in the negative a question posed by Kleinbock, Lindenstrauss, and Weiss, by constructing a family …

三维Veronese曲线上的联立丢番图逼近：完整的豪斯多夫维数结论

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。

On inhomogeneous Diophantine approximation and Hausdorff dimension

Let $Γ= Z A +Z^n$ be a dense subgroup with rank $n+1$ in $R^n$ and let $ω(A)$ denote the exponent of uniform simultaneous rational approximation to the point $A$. We show that for any real number $v\ge ω(A)$, the Hausdorff dimension of the set $B_v$ of points in $R^n$ which are $v$-approximable with respect to $Γ$, is equal to $1/v$.

Diophantine equations defined by binary quadratic forms over rational function fields

We study the ``imaginary" binary quadratic form equations ax^2+bxy+cy^2+g=0 over k[t] in rational function fields, showing that a condition with respect to the Artin reciprocity map, is the only obstruction to the local-global principle for integral solutions of the equation.

素数丢番图逼近与Landau–Siegel零点

Diophantine approximation by primes and Landau--Siegel zeros

本文在假设Siegel零点存在或足够强的条件下，证明了有限型无理数$\alpha$与实数$\beta$的线性组合$\alpha p+\beta$可被素数$p$逼近到$p^{-1/3+\varepsilon}$或$p^{-1/3-1/20}$的精度，给出了素数丢番图逼近的新结果。

On a mixed problem in Diophantine approximation

Let $d$ be a positive integer. Let $p$ be a prime number. Let $α$ be a real algebraic number of degree $d+1$. We establish that there exist a positive constant $c$ and infinitely many algebraic numbers $ξ$ of degree $d$ such that $|α- ξ| \cdot \min\{|\Norm(ξ)|_p,1\} < c H(ξ)^{-d-1} (\log 3 H(ξ))^{-1/d}$. Here, $H(ξ)$ and $\Norm(ξ)$ denote the na{\"ı}ve height of $ξ$ and its norm, respectively. This extends an earlier result of de Mathan and Teulié that deals with the case $d=1$.

Diophantine exponents of measures: a dynamical approach

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.

Optimality of two inequalities for exponents of Diophantine approximation

We investigate two inequalities of Bugeaud and Laurent, each involving triples of classical exponents of Diophantine approximation associated to $\ux\in\mathbb{R}^n$. We provide a complete description of parameter triples that admit equality for suitable $\ux$, which turns out rather surprising. For $n=2$ our results agree with work of Laurent. Moreover, we establish lower bounds for the Hausdorff and packing dimensions of the involved $\ux$, and in special cases we can show they are sharp. Proofs are based on the variational principle in parametric geometry of numbers, we enclose sketches of …

Is the quartic Diophantine equation $A^4+hB^4=C^4+hD^4$ solvable for any integer $h$?

The Diophantine equation $A^4+hB^4=C^4+hD^4$, where $h$ is a fixed arbitrary positive integer, has been investigated by some authors. Currently, by computer search, the integer solutions of this equation are known for all positive integer values of $h \le 5000$ and $A, B, C, D \le 100000$, except for some numbers, while a solution of this Diophantine equation is not known for arbitrary positive integer values of $h$. Gerardin and Piezas found solutions of this equation when $h$ is given by polynomials of degrees $5$ and $2$ respectively. Also Choudhry presented some new solutions of this equat…

Khintchine's theorem and transference principle for star bodies

Analogues of Khintchine's Theorem in simultaneous Diophantine approximation in the plane are proved with the classical height replaced by fairly general planar distance functions or equivalently star bodies. Khintchine's transference principle is discussed for distance functions and a direct proof for the multiplicative version is given. A transference principle is also established for a different distance function.

Equidistribution on homogeneous spaces and the distribution of approximates in Diophantine approximation

The present paper is concerned with equidistribution results for certain flows on homogeneous spaces and related questions in Diophantine approximation. Firstly, we answer in the affirmative, a question raised by Kleinbock, Shi and Weiss regarding equidistribution of orbits of arbitrary lattices under diagonal flows and with respect to unbounded functions. We then consider the problem of Diophantine approximation with respect to rationals in a fixed number field. We prove a number field analogue of a famous result of W. M.Schmidt which counts the number of approximates to Diophantine inequalit…

An equivalence principle between polynomial and simultaneous Diophantine approximation

We show that Mahler's classification of real numbers $ζ$ with respect to the growth of the sequence $(w_{n}(ζ))_{n\geq 1}$ is equivalently induced by certain natural assumptions on the decay of the sequence $(λ_{n}(ζ))_{n\geq 1}$ concerning simultaneous rational approximation. Thereby we obtain a much clearer picture on simultaneous approximation to successive powers of a real number in general. Another variant of the Mahler classification concerning uniform approximation by algebraic numbers is shown as well. Our method has several applications to classic exponents of Diophantine approximatio…

Diophantine approximation with constraints

Following Schmidt, Thurnheer and Bugeaud-Kristensen, we study how Dirichlet's theorem on linear forms needs to be modified when one requires that the vectors of coefficients of the linear forms make a bounded acute angle with respect to a fixed proper non-zero subspace $V$ of $\mathbb{R}^n$. Assuming that the point of $\mathbb{R}^n$ that we are approximating has linearly independent coordinates over $\mathbb{Q}$, we obtain best possible exponents of approximation which surprisingly depend only on the dimension of $V$. Our estimates are derived by reduction to a result of Thurnheer, while their…

Solution of Cassels' Problem on a Diophantine Constant over Function Fields

This paper deals with the analogue of Inhomogeneous Diophantine Approximation in function fields. The inhomogeneous approximation constant of a Laurent series $θ\in\mathbb F_q\left(\left(\frac{1}{t}\right)\right)$ with respect to $γ\in\mathbb F_q\left(\left(\frac{1}{t}\right)\right)$ is defined to be $c(θ,γ)=\inf_{0\neq N\in\mathbb F_q\left[t\right]}|N|\cdot|\langle Nθ- γ\rangle|$. We show that for every $θ$ there exists $γ$ such that $c(θ,γ)\geq q^{-2}$, and find a sufficient condition on $θ$ which forces $c(θ,γ) \leq q^{-2}$ for every $γ$. Given $θ$, we prove that the set $BA_θ=\left\{γ\in\m…

Modified Schmidt games and Diophantine approximation with weights

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.

过度性、无理性与丢番图逼近

Extravagance, irrationality and Diophantine approximation

该研究探讨高斯映射下测度与丢番图逼近的关系，证明部分商对数不可积的连分数混合测度下几乎所有数为刘维尔数，构造任意无理性指数的高斯不变遍历测度，并给出弱Renyi测度下的辛钦型二分法。

Diophantine Approximation on projective Varieties I: Algebraic distance and metric Bézout Theorem

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.

关于具有丢番图频率的拟周期函数用周期函数逼近的研究

On the approximation of quasiperiodic functions with Diophantine frequencies by periodic functions

本文分析具有丢番图频率的拟周期函数用周期函数逼近的误差，给出误差界与逼近速率，并通过实例验证。

Inhomogeneous Diophantine approximation for generic homogeneous functions

The present paper is a sequel to [Monatsh.~Math.\ {\bf 194} (2021), 523--554] in which results of that paper are generalized so that they hold in the setting of inhomogeneous Diophantine approximation. Given any integers $n \geq 2$ and $\ell \geq 1$, any ${\pmb ξ} = \left(ξ_1, \dots , ξ_\ell \right) \in \mathbb{R}^\ell$, and any homogeneous function \linebreak $f = \left(f_1, \dots , f_\ell \right): \mathbb{R}^n \to \mathbb{R}^\ell$ that satisfies a certain nonsingularity assumption, we obtain a biconditional criterion on the approximating function $ψ= \left(ψ_1, \dots , ψ_\ell \right): \mathb…

Tiling of rectangles with squares via Diophantine approximation

This article shines new light on the classical problem of tiling rectangles with squares efficiently with a novel method. With a twist on the traditional approach of resistor networks, we provide new and improved results on the matter using the theory of Diophantine Approximation, hence overcoming long-established difficulties, such as generalizations to higher-dimensional analogues. The universality of the method is demonstrated through its applications to different tiling problems. These include tiling rectangles with other rectangles, with their respective higher-dimensional counterparts, a…

关于丢番图集类的和集与笛卡尔积的度量结果

Metric results on sumsets and Cartesian products of classes of Diophantine sets

受Erdős关于刘维尔数和积结果启发，研究指定无理性指数实数类和集的度量性质，证明其和集勒贝格测度下几乎覆盖全体实数，补集豪斯多夫维数小，还拓展到笛卡尔积维数、缺数字康托尔集填充维数等问题，推广了Erdős的方法。

On the Exponential Diophantine Equation $(a^2-2)(b^2-2)=x^2$

In this paper, we consider the equation $(a^n-2^{m})(b^n-2^{m})=x^2$. By assuming the abc conjecture is true, in [8], Luca and Walsh gave a theorem, which implies that the above equation has only finitely many solutions $n,x$ if a and b are different fixed positive integers. We solve the above equation when $m=1$ and $(a,b)=(2,10),(4,100),(10,58),(3,45)$. Moreover, we show that $(a^2-2)(b^2-2)=x^2$ has no solution n,x if 2|n and gcd$(a,b)=1$. We also give a conjecture which says that the equation $(2^2-2)((2P_k)^n-2)=x^2$ has only the solution $(n,x)=(2,Q_k)$, where $k>3$ is odd and $P_k,Q_k$ …

Uniformly de Bruijn sequences and symbolic Diophantine approximation on fractals

Intrinsic Diophantine approximation on fractals, such as the Cantor ternary set, was undoubtedly motivated by questions asked by K. Mahler (1984). One of the main goals of this paper is to develop and utilize the theory of infinite de Bruijn sequences in order to answer closely related questions. In particular, we prove that the set of infinite de Bruijn sequences in $k\geq 2$ letters, thought of as a set of real numbers via a decimal expansion, has positive Hausdorff dimension. For a given $k$, these sequences bear a strong connection to Diophantine approximation on certain fractals. In parti…

Inside factorial monoids and the cale monoid of a single Diophantine equation

We give a structure theorem for inside factorial domains. As an example we study the monoid of nonnegative integer solutions of equations of the form $a_1x_1+\cdots +a_{r-1}x_{r-1}=a_rx_r$, with $a_1,\ldots,a_r$ positive integers. This set is isomorphic to a simplicial full affine semigroup, and thus it can be described in terms of its extremal rays and the Apéry sets with respect to the extremal rays.

Intrinsic Diophantine Approximation for overlapping iterated function systems

In this paper we study a family of limsup sets that are defined using iterated function systems. Our main result is an analogue of Khintchine's theorem for these sets. We then apply this result to the topic of intrinsic Diophantine Approximation on self-similar sets. In particular, we define a new height function for an element of $\mathbb{Q}^d$ contained in a self-similar set in terms of its eventually periodic representations. For limsup sets defined with respect to this height function, we obtain a detailed description of their metric properties. The results of this paper hold in arbitrary …

Geodesic cusp excursions and metric diophantine approximation

We derive several results that describe the rate at which a generic geodesic makes excursions into and out of a cusp on a finite area hyperbolic surface and relate them to approximation with respect to the orbit of infinity for an associated Fuchsian group. This provides proofs of some well known theorems from metric diophantine approximation in the context of Fuchsian groups. It also gives new results in the classical setting.

齐性空间上的非扩张随机游走与丢番图逼近

Non-Expanding Random walks on Homogeneous spaces and Diophantine approximation

本文研究仿射格空间上的非扩张随机游走，建立平稳测度分类定理，并将其通用性与Birkhoff通用性关联，进而应用于分形上的非齐次丢番图逼近问题。

Diophantine approximations on fractals

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 …

Seshadri constants, Diophantine approximation, and Roth's Theorem for arbitrary varieties

In this paper, we associate an invariant $α_{x}(L)$ to an algebraic point $x$ on an algebraic variety $X$ with an ample line bundle $L$. The invariant $α$ measures how well $x$ can be approximated by rational points on $X$, with respect to the height function associated to $L$. We show that this invariant is closely related to the Seshadri constant $ε_{x}(L)$ measuring local positivity of $L$ at $x$, and in particular that Roth's theorem on $\mathbf{P}^1$ generalizes as an inequality between these two invariants valid for arbitrary projective varieties.

Hausdorff测度和Fourier维度在加权和乘法Diophantine逼近中出现的limsup集

Hausdorff measure and Fourier dimensions of limsup sets arising in weighted and multiplicative Diophantine approximation

本文研究了加权和乘法Diophantine逼近中出现的limsup集的Lebesgue测度、Hausdorff测度和Fourier维度，提供了零-全律判定准则，并确定了这些集的精确Fourier维度。

加权扭曲非齐次丢番图逼近

Weighted twisted inhomogeneous Diophantine approximation

本研究在非齐次丢番图逼近领域，利用加权遍在性与加权质量转移原理，证明了Kurzweil定理的多维加权版本，给出了对应点集的勒贝格测度零满律与豪斯多夫维数结果。

The hyperbolic geometry of Markov's theorem on Diophantine approximation and quadratic forms

Markov's theorem classifies the worst irrational numbers with respect to rational approximation and the indefinite binary quadratic forms whose values for integer arguments stay farthest away from zero. The main purpose of this paper is to present a new proof of Markov's theorem using hyperbolic geometry. The main ingredients are a dictionary to translate between hyperbolic geometry and algebra/number theory, and some very basic tools borrowed from modern geometric Teichmüller theory. Simple closed geodesics and ideal triangulations of the modular torus play an important role, and so do the pr…

Weighted uniform Diophantine approximation of systems of linear forms

Following the development of weighted asymptotic approximation properties of matrices, we introduce the analogous uniform approximation properties (that is, study the improvability of Dirichlet's Theorem). An added feature is the use of general norms, rather than the supremum norm, to quantify the approximation. In terms of homogeneous dynamics, the approximation properties of an $m \times n$ matrix are governed by a trajectory in $\mathrm{SL}_{m+n}({\mathbb R})/\mathrm{SL}_{m+n}({\mathbb Z})$ avoiding a compact subset of the space of lattices called the critical locus defined with respect to …

环面矩阵变换的扭曲丢番图逼近

Twisted Diophantine approximation for matrix transformations of tori

本文研究环面矩阵变换下的扭曲丢番图逼近问题，针对不同测度情形证明了对应逼近集合的度量二分性，解决了相关猜想变体并建立Jarník型定理。

Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces

We study Lagrange spectra at cusps of finite area Riemann surfaces. These spectra are penetration spectra that describe the asymptotic depths of penetration of geodesics in the cusps. Their study is in particular motivated by Diophantine approximation on Fuchsian groups. In the classical case of the modular surface and classical Diophantine approximation, Hall proved in 1947 that the classical Lagrange spectrum contains a half-line, known as a Hall ray. We generalize this result to the context of Riemann surfaces with cusps and Diophantine approximation on Fuchsian groups. One can measure excu…

齐性空间上拟射线的有界轨道与带权函数的丢番图逼近

Bounded trajectories of quasi-rays on homogeneous spaces and Diophantine approximation with weight functions

研究齐性空间上拟射线有界轨道，通过施氏等分布定理描述\(G\)中一类子集\(F\)，使其有界轨道点集有全豪斯多夫维数，还证明了拟乘法权函数丢番图逼近中不良逼近矩阵集有全豪斯多夫维数。

关于定量子空间定理的定性方面

On qualitative aspects of the quantitative subspace theorem

本文基于参数化子空间定理，通过引入线性截面和对数扭曲高度函数等工具，推导了数域有限扩张下大线性系统的丢番图算术不等式，并建立了丢番图例外集的新颖线性散射型结果。