Multiplicative Diophantine Exponents of Hyperplanes

Asymptotic Diophantine approximation: The multiplicative case

Let $α$ and $β$ be irrational real numbers and $0<\F<1/30$. We prove a precise estimate for the number of positive integers $q\leq Q$ that satisfy $\|qα\|\cdot\|qβ\|<\F$. If we choose $\F$ as a function of $Q$ we get asymptotics as $Q$ gets large, provided $\F Q$ grows quickly enough in terms of the (multiplicative) Diophantine type of $(α,β)$, e.g., if $(α,β)$ is a counterexample to Littlewood's conjecture then we only need that $\F Q$ tends to infinity. Our result yields a new upper bound on sums of reciprocals of products of fractional parts, and sheds some light on a recent question of Lê …

Fully Inhomogeneous Multiplicative Diophantine Approximation of Badly Approximable Numbers

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.

Sums of reciprocals of fractional parts and multiplicative Diophantine approximation

There are two main interrelated goals of this paper. Firstly we investigate the sums \[ S_N(α,γ):=\sum_{n=1}^N\frac{1}{n\|nα-γ\|}~~~\text{and}~~~ R_N(α,γ):=\sum_{n=1}^N\frac{1}{\|nα-γ\|}\,, \] where $α$ and $γ$ are real parameters and $\|\cdot\|$ is the distance to the nearest integer. Our theorems improve upon previous results of W. M. Schmidt and others, and are (up to constants) best possible. Related to the above sums, we also obtain upper and lower bounds for the cardinality of \[ \{1\le n\le N:\|nα-γ\|<\varepsilon\} \, , \] valid for all sufficiently large $N$ and all sufficiently small …

Multiplicative $p$-adic metric Diophantine approximation on manifolds and dichotomy of exponents

In this paper we study $p$-adic Diophantine approximation on manifolds, specifically multiplicative Diophantine approximation on affine subspaces and a Diophantine dichotomy for analytic $p$-adic manifolds.

移位乘法子群的乘法结构及其在丢番图元组中的应用

Multiplicative structure of shifted multiplicative subgroups and its applications to Diophantine tuples

研究移位乘法子群的乘法结构及其与加性组合学和丢番图方程理论的联系，提出新的结果，包括改进的上界和对Sárközy猜想的进展。

Higher-rank Bohr sets and multiplicative diophantine approximation

Gallagher's theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation. In addition, we generalise the theory and result to the inhomogeneous setting. …

乘法矩阵方程的可判定性及相关丢番图问题

Decidability of multiplicative matrix equations and related Diophantine problems

本文研究了代数数域上乘法矩阵方程的可判定性，通过有效方法分析丢番图问题，提出了解的大小和数量的显式界，并探讨了对称矩阵系统的相关计数问题。

Convergence results for simultaneous and multiplicative Diophantine approximation on planar curves

Let $\mathcal{C}$ be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation with two independent approximation functions; that is if a certain sum converges then the set of all points $(x,y)$ on the curve which satisfy simultaneously the inequalities $\| q x \| < ψ_1(q)$ and $\| qy \| < ψ_2(q)$ infinitely often has induced measure 0. This completes the metric theory for the Lebesgue case. Further, for the cae of multiplicative approximation $\| qx \| \| q y \| < ψ(q)$, we establish a Hausdorff measure convergence re…

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。

缺失数字分形上的联立与乘法丢番图逼近

Simultaneous and multiplicative Diophantine approximation on missing-digit fractals

本文研究缺失数字分形上的丢番图逼近度量理论，建立了欣钦定理和加拉格尔定理的类比及非齐次推广。

超几何区域中的有理点与流形上的乘法丢番图逼近

Rational Points in Hyperbolic Regions and Multiplicative Diophantine Approximation on Manifolds

该研究解决了非退化流形和仿射子空间的乘法丢番图逼近收敛理论问题，改进了相关强极值性结果。

There is no analogue to Jarník's relation for twisted Diophantine approximmation

Jarník's relation is in dimension $2$ the formula $\hatλ + \frac{1}{\hatω} = 1$ linking both uniform exponents. It is an open question to generalize this equation to higher dimension, or to the multiplicative case. In this paper we consider a twisted case, between the classical and the multiplicative one, and we show that no analogue to Jarník's relation holds.

乘法丢番图逼近中的非Salem集

Non-Salem sets in multiplicative Diophantine approximation

本文回答了Cai-Hambrook的问题，计算了乘法ψ-良逼近集的傅里叶维数，并证明对τ>1，集合M_2^×(q↦q^{-τ})是非Salem的。

乘法丢番图逼近中的中心极限定理

Central Limit Theorems in Multiplicative Diophantine Approximation

本文研究了乘法丢番图逼近问题中的整数解数量，通过分析同质空间上的测度相关性及Siegel变换估计，证明了计数函数在分布上收敛于正态分布。

Diophantine approximation with arithmetic functions, I

We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.

膨胀空隙序列的分布及其在乘法丢番图逼近中的应用

The dispersion of dilated lacunary sequences, with applications in multiplicative Diophantine approximation

本文研究 Hadamard 空隙序列膨胀集模 1 的最大间隙，证明存在膨胀因子使间隙阶至多为 (log N)/N，且对几乎所有的膨胀因子该阶至多为 (log N)^{2+ε}/N，并由此改进乘法丢番图逼近中 Littlewood 问题非齐次版本的上界。

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。

On the metric theory of multiplicative Diophantine approximation

In 1962, Gallagher proved an higher dimensional version of Khintchine's theorem on Diophantine approximation. Gallagher's theorem states that for any non-increasing approximation function $ψ:\mathbb{N}\to (0,1/2)$ with $\sum_{q=1}^{\infty} ψ(q)\log q=\infty$ and $γ=γ'=0$ the following set \[ \{(x,y)\in [0,1]^2: \|qx-γ\|\|qy-γ'\|<ψ(q) \text{ infinitely often}\} \] has full Lebesgue measure. Recently, Chow and Technau proved a fully inhomogeneous version (without restrictions on $γ,γ'$) of the above result. In this paper, we prove an Erdős-Vaaler type result for fibred multiplicative Diophantine…

带限制分母的乘性丢番图逼近

Multiplicative Diophantine approximation with restricted denominators

本文研究了带限制分母的乘性丢番图逼近问题，完全确定了满足特定不等式无限次的点集的Hausdorff维数，并基于Hausdorff测度的一般收敛性结果得出结论。

Bohr sets and multiplicative diophantine approximation

In two dimensions, Gallagher's theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. We prove an inhomogeneous fibre version of Gallagher's theorem, sharpening and making unconditional a result recently obtained conditionally by Beresnevich, Haynes and Velani. The idea is to find large generalised arithmetic progressions within inhomogeneous Bohr sets, extending a construction given by Tao. This precise structure enables us to verify the hypotheses of the Duffin--Schaeffer theorem for the problem at hand, via the geometry of numbers.

丢番图指数的几何

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。

$\mathbb{R}^d$-作用的子流形一般性与一致乘性丢番图逼近

Submanifold-genericity of $\mathbb{R}^d$-actions and uniform multiplicative Diophantine approximation

本文为$\mathbb{R}^d$-作用证明了涉及膨胀子流形平均的新遍历定理，给出有效混合假设下的误差项定量估计，并据此研究了实系数矩阵的乘性Dirichlet可改进性，证明了几乎所有矩阵的一致逼近性，提出了Littlewood猜想的强化问题。

丢番图逼近中的Littlewood与Duffin–Schaeffer型问题

Littlewood and Duffin--Schaeffer-type problems in diophantine approximation

该研究在丢番图逼近领域，建立了Gallagher定理的完全非齐次版本、丢番图纤维细化结果及Liouville纤维的尖锐阈值，还证明了一类非单调逼近函数下Duffin–Schaeffer猜想的非齐次版本。

Multiplicative parametric geometry of numbers and transference theorems for lattice exponents

In this paper we adapt parametric geometry of numbers developed by Wolfgang Schmidt and Leonard Summerer to a multiplicative setting, and derive a chain of inequalities for the corresponding exponents which splits the transference inequality for Diophantine exponents of lattices in the same way Khintchine's transference inequalities for simultaneous approximation can be split.

直线上乘法丢番图逼近的有效等分布

Effective equidistribution for multiplicative Diophantine approximation on lines

本文通过证明 SL(3,R)/SL(3,Z) 中幂幺轨道的有效等分布，将 Littlewood 猜想在直线上几乎处处加强了两个对数因子，并利用对偶 Bohr 集理论改进 Kleinbock 的收敛结果及齐性空间对数律。

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…

Counting rational points close to $p$-adic integers and applications in Diophantine approximation

We find upper and lower bounds on the number of rational points that are $ψ$-approximations of some $n$-dimensional $p$-adic integer. Lattice point counting techniques are used to find the upper bound result, and a Pigeon-hole principle style argument is used to find the lower bound result. We use these results to find the Hausdorff dimension for the set of $p$-adic weighted simultaneously approximable points intersected with $p$-adic coordinate hyperplanes. For the lower bound result we show that the set of rational points that $τ$-approximate a $p$-adic integer form a set of resonant points …

Diophantine approximations on random fractals

We show that fractal percolation sets in $\mathbb{R}^{d}$ almost surely intersect every hyperplane absolutely winning (HAW) set with full Hausdorff dimension. In particular, if $E\subset\mathbb{R}^{d}$ is a realization of a fractal percolation process, then almost surely (conditioned on $E\neq\emptyset$), for every countable collection $\left(f_{i}\right)_{i\in\mathbb{N}}$ of $C^{1}$ diffeomorphisms of $\mathbb{R}^{d}$, $\dim_{H}\left(E\cap\left(\bigcap_{i\in\mathbb{N}}f_{i}\left(\text{BA}_{d}\right)\right)\right)=\dim_{H}\left(E\right)$, where $\text{BA}_{d}$ is the set of badly approximable …

乘法丢番图逼近的Duffin-Schaeffer猜想

The Duffin-Schaeffer Conjecture for multiplicative Diophantine approximation

本文证明了乘法丢番图逼近的Duffin-Schaeffer型结果，去除了单调性假设，给出了无穷多个逼近成立的充要条件，解决了Beresnevich、Haynes和Velani的猜想。

流形中坏点的超平面获胜性质

Hyperplane winning for bad in manifolds

该研究证明了非退化解析流形及满足几乎最优丢番图条件的仿射子空间内加权坏逼近向量的超平面获胜性质，回答了 Badziahin–Velani 关于高维流形的猜想，拓展了此前仅在二维空间的相关结果。

Multiplicative zero-one laws and metric number theory

We develop the classical theory of Diophantine approximation without assuming monotonicity or convexity. A complete `multiplicative' zero-one law is established akin to the `simultaneous' zero-one laws of Cassels and Gallagher. As a consequence we are able to establish the analogue of the Duffin-Schaeffer theorem within the multiplicative setup. The key ingredient is the rather simple but nevertheless versatile `cross fibering principle'. In a nutshell it enables us to `lift' zero-one laws to higher dimensions.

On a mixed Khintchine problem in Diophantine approximation

We establish a `mixed' version of a fundamental theorem of Khintchine within the field of simultaneous Diophantine approximation. Via the notion of ubiquity we are able to make significant progress towards the completion of the metric theory associated with mixed problems in this setting. This includes finding a natural mixed analogue of the classical Jarník-Besicovich Theorem. Previous knowledge surrounding mixed problems was almost entirely restricted to the multiplicative setup of de Mathan & Teulié [21], where the concept originated.

乘性丢番图逼近中的一个一致度量定理

A uniform metrical theorem in multiplicative Diophantine approximation

本文研究了勒贝格通用二维实数对的乘性丢番图逼近问题，给出了满足特定增长条件下相关小值乘积数量的渐近公式。

Multiplicative relations among singular moduli

We consider some Diophantine problems of mixed modular-multiplicative type associated with the Zilber-Pink conjecture. In particular, we prove a finiteness statement for the number of multiplicative relations between singular moduli (j-invariants of elliptic curves with complex multiplication.)

Mass transference principle from rectangles to rectangles in Diophantine approximation

By introducing a ubiquity property for rectangles, we prove the mass transference principle from rectangles to rectangles, i.e., if a sequence of rectangles forms a ubiquity system (a full measure property), then the limsup set defined by shrinking these rectangles to smaller rectangles has full Hausdorff measure or to say transfer a full measure property to a full Hausdorff measure property for brevity. The limsup sets generated by balls or generated by rectangles appear at the most fundamental level in Diophantine approximation: one follows from Dirichlet's theorem, the other follows from Mi…

Simultaneous Diophantine approximation - logarithmic improvements

This paper is devoted to the study of a problem of Cassels in multiplicative Diophantine approximation which involves minimising values of a product of affine linear forms computed at integral points. It was previously known that values of this product become arbitrary close to zero, and we establish that, in fact, they approximate zero with an explicit rate. Our approach is based on investigating quantitative density of orbits of higher-rank abelian groups.

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 $ \…

指数丢番图逼近与符号动力学

Exponential Diophantine approximation and symbolic dynamics

本文扩展了交织乘性Markoff-Lagrange谱与符号动力学的关键公式，运用复分析进行证明，并将新方法推广至可能含重根的多项式，导出了谱上的若干预期推论。

Extrinsic Diophantine approximation on manifolds and fractals

Fix $d\in\mathbb N$, and let $S\subseteq\mathbb R^d$ be either a real-analytic manifold or the limit set of an iterated function system (for example, $S$ could be the Cantor set or the von Koch snowflake). An $extrinsic$ Diophantine approximation to a point $\mathbf x\in S$ is a rational point $\mathbf p/q$ close to $\mathbf x$ which lies $outside$ of $S$. These approximations correspond to a question asked by K. Mahler ('84) regarding the Cantor set. Our main result is an extrinsic analogue of Dirichlet's theorem. Specifically, we prove that if $S$ does not contain a line segment, then for ev…

Diophantine approximation with nonsingular integral transformations

Let $Γ$ be the multiplicative semigroup of all $n\times n$ matrices with integral entries and positive determinant. Let $1\leq p \leq n-1$ and $V=\R^n\oplus \cdots \oplus \R^n$ ($p$ copies). We consider the componentwise action of $Γ$ on $V$. Let $\bx\in V$ be such that $Γ\bx$ is dense in $V$. We discuss the effectiveness of the approximation of any target point $\by \in V$ by the orbit $\{ γ\bx \mid γ\in Γ\}$, in terms of $\norm γ\norm$, and prove in particular that for all $\bx$ in the complement of a specific null set described in terms of a certain Diophantine condition, the exponent of ap…

算术和乐界与有效丢番图逼近

Arithmetic holonomy bounds and effective Diophantine approximation

本研究通过量化算术和乐界提出射影直线与乘法群上有效丢番图逼近新方法，推导出代数数高阶根的有效无理性度量，可导出经典有效丢番图定理并给出多个数论应用。

带权的 manifold 上的丢番图逼近

Weighted Diophantine approximation on manifolds

本文在流形上建立了带权的同时Khinchine型定理，并扩展了先前结果，同时得到了乘法Khinchine型收敛定理。

余切和与丢番图和的极限定律

Limit laws for cotangent and Diophantine sums

本文针对带幂奇点的1周期函数相关和式，证明其极限定律并给出收敛速度估计，结果适用于余切和与分数部分倒数和，主要工具包括Schmidt方法、Gauss-Kuzmin问题及$ψ$-混合随机变量理论。

逼近元的计数与联合等分布

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。

Spherical averages of Siegel transforms for higher rank diagonal actions and applications

We investigate the geometry of approximates in multiplicative Diophantine approximation. Our main tool is a new multiparameter averaging result for Siegel transforms on the space of unimodular lattices in ${\mathbb R}^n$ which is of independent interest.

Expanding translates of curves and Dirichlet-Minkowski theorem on linear forms

We show that a multiplicative form of Dirichlet's theorem on simultaneous Diophantine approximation as formulated by Minkowski, cannot be improved for almost all points on any analytic curve on R^k which is not contained in a proper affine subspace. Such an investigation was initiated by Davenport and Schmidt in the late sixties. The Diophantine problem is then settled by showing that certain sequence of expanding translates of curves on the homogeneous space of unimodular lattices in R^{k+1} gets equidistributed in the limit. We use Ratner's theorem on unipotent flows, linearization technique…

扭曲的丢番图逼近的素数时间

The Prime times of twisted Diophantine approximation

本文研究了在特定乘法结构的数集上，坏逼近数的不等式解集性质，改进了Khintchine型结果，并通过构造非平凡集证明了坏逼近假设的必要性。

逼近向量的几何与算术性质

Geometric and arithmetic aspects of approximation vectors

该研究探讨丢番图逼近中与逼近向量相关的三类几何、算术量的渐近分布，通过阿代尔空间单参数流截面等分布方法，得到几乎所有实数向量及全实数域情形下的极限测度，推广了前人结果至高维与联合等分布场景。