Yuqing Zhang
We apply nondivergence estimates for flows on homogeneous spaces to compute Diophantine exponents of affine subspaces of $\R^n$ and their nondegenerate submanifolds.
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We show that affine coordinate subspaces of dimension at least two in Euclidean space are of Khintchine type for divergence. For affine coordinate subspaces of dimension one, we prove a result which depends on the dual Diophantine type of the base point of the subspace. These results provide evidence for the conjecture that all affine subspaces of Euclidean space are of Khintchine type for divergence. We also prove a partial analogue regarding the Hausdorff measure theory. Furthermore, we obtain various results relating weighted Diophantine approximation and Dirichlet improvability. In particu…
Let $F \in \mathbb{Z}[x_1, \ldots, x_n]$ be a homogeneous form of degree $d \geq 2$, and let $V_F^*$ denote the singular locus of the affine variety $V(F) = \{ \mathbf{z} \in {\mathbb{C}}^n: F(\mathbf{z}) = 0 \}$. In this paper, we prove the existence of integer solutions with prime coordinates to the equation $F(x_1, \ldots, x_n) = 0$ provided $F$ satisfies suitable local conditions and $n - \dim V_F^* \geq 2^8 3^4 5^2 d^3 (2d-1)^2 4^{d}$. Our result improves on what was known previously due to Cook and Magyar (B. Cook and A. Magyar, `Diophantine equations in the primes'. Invent. Math. 198 (2…
Recently, Adiceam, Beresnevich, Levesley, Velani and Zorin proved a quantitative version of the convergence case of the Khintchine-Groshev theorem for nondegenerate manifolds, motivated by applications to interference alignment. In the present paper, we obtain analogues of their results for affine subspaces.
We prove the convergence and divergence cases of an inhomogeneous Khintchine-Groshev type theorem for dual approximation restricted to affine subspaces in $\mathbb{R} ^n$. The divergence results are proved in the more general context of Hausdorff measures.
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.
We study the problem of Diophantine approximation on lines in $\mathbb{R}^d$ under certain primality restrictions.
We prove a new lower bound for the exponent of growth of the best two-dimensional Diophantine approximations with respect to Euclidean norm.
Answering two questions of Beresnevich and Velani, we develop zero-one laws in both simultaneous and multiplicative Diophantine approximation. Our proofs rely on a Cassels-Gallagher type theorem as well as a higher-dimensional analogue of the cross fibering principle of Beresnevich, Haynes and Velani.
We investigate the problem of best simultaneous Diophantine approximation under a constraint on the denominator, as proposed by Jurkat. New lower estimates for optimal approximation constants are given in terms of critical determinants of suitable star bodies. Tools are results on simultaneous Diophantine approximation of rationals by rationals with smaller denominator. Finally, the approximation results are applied to the decomposition of integer vectors.
We compute the sequence of best Diophantine approximations for some pairs of cubic Pisot numbers which do not satisfy the Property (F).
In this paper we prove an inequality for individual and uniform Diophantine exponents in the case of simultaneous approximation. This inequality is better than Jarnik's for small values of the uniform exponent.
We investigate the Diophantine property of a pair of elements in the group of affine transformations of the line. We say that a pair of elements g_1,g_2 in this group is Diophantine if there is a number A such that a product of length l of elements of the set {g_1,g_2,g_1^{-1},g_2^{-1}} is either the unit element or of distance at least A^{-l} from the unit element. We prove that the set of non-Diophantine pairs in a certain one parameter family is of Hausdorff dimension 0.
A note on the quality of simultaneous Diophantine approximations obtained by the LLL algorithm
本文针对LLL算法所得联立丢番图逼近的质量问题，对Bosma和Smeets的相关结果作出了显著改进。
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…
We give an algorithm that finds a sequence of approximations with Dirichlet coefficients bounded by a constant only depending on the dimension. The algorithm uses the LLL-algorithm for lattice basis reduction. We present a version of the algorithm that runs in polynomial time of the input.
In this paper, we solve the simultaneous Diophantine equations m.(x_1^k+....+x_{t_1}^k)=n.(y_1^k+....+y_{t_2}^k); k=1,3, where t_1, t_2>3, and m, n are fixed arbitrary and relatively prime positive integers. This is done by choosing two appropriate trivial parametric solutions and obtaining infinitely many nontrivial parametric solutions. Also we work out some examples, in particular the Diophantine systems of A^k+B^k+C^k=D^k+E^4; k=1,3.
The goal of this paper is to develop the theory of weighted Diophantine approximation of rational numbers to $p$-adic numbers. Firstly, we establish complete analogues of Khintchine's theorem, the Duffin-Schaeffer theorem and the Jarník-Besicovitch theorem for `weighted' simultaneous Diophantine approximation in the $p$-adic case. Secondly, we obtain a lower bound for the Hausdorff dimension of weighted simultaneously approximable points lying on $p$-adic manifolds. This is valid for very general classes of curves and manifolds and have natural constraints on the exponents of approximation. Th…
Let Q be an infinite set of positive integers. Denote by W(Q) the set of n-tuples of real numbers simultaneously tau-well approximable by infinitely many rationals with denominators in Q but only by finitely many rationals with denominators in the complement of Q. The Hausdorff dimension of the liminf set W(Q) is computed when tau > 2 + 1/n. A p-adic analogue of the problem is also studied.
Exact Diophantine approximation$\colon$ the simultaneous case in $\mathbb{R}^{2}$
该研究填补了Fregoli关于$\mathbb{R}^{2}$中精确逼近阶集合豪斯多夫维数研究的空白，完善了联立情形下精确丢番图逼近的相关理论。
The paper is devoted to the problem of estimating the constant of the best Diophantine approximations. The estimates of lower bound $ C_n $ for $ n = 5 $ and $ n = 6 $ was improved. The first chapter gives an overview of the history of estimates of the constant of the best Diophantine approximations. In the second chapter sets out the methods and results of numerical experiments leading to estimates of $ C_n $. The third chapter is devoted to estimating some functions by means of which in the fourth chapter we will proof the estimates of $ C_n $ for $ 3 \leq n \leq 6 $.
For any given real number $α$ with bounded partial quotients, we construct explicitly continuum many real numbers $β$ with bounded partial quotients for which the pair $(α, β)$ satisfies a strong form of the Littlewood conjecture. Our proof is elementary and rests on the basic theory of continued fractions.
Simultaneous Diophantine approximation with a divisibility condition
本文研究了 ${\bf R}^2$ 中与连分数类似的联立有理逼近序列，证明在添加可除性条件后，先前关于 Littlewood 猜想的相关结果依然成立。
We present a hypergeometric construction of rational approximations to $ζ(2)$ and $ζ(3)$ which allows one to demonstrate simultaneously the irrationality of each of the zeta values, as well as to estimate from below certain linear forms in 1, $ζ(2)$ and $ζ(3)$ with rational coefficients. A new notion of (simultaneous) diophantine exponent is introduced to formalise the arithmetic structure of these specific linear forms. Finally, the properties of this newer concept are studied and linked to the classical irrationality exponent and its generalisations given recently by S. Fischler.
We give several upper bounds for the uniform simultaneous Diophantine exponent $\widehatλ_n(ξ)$ of a transcendental number $ξ\in\mathbb{R}$. The most important one relates $\widehatλ_n(ξ)$ and the ordinary simultaneous exponent $ω_k(ξ)$ in the case when $k$ is substantially smaller than $n$. In particular, in the generic case $ω_k(ξ)=k$ with a properly chosen $k$, the upper bound for $\widehatλ_n(ξ)$ becomes as small as $\frac{3}{2n} + O(n^{-2})$ which is substantially better than the best currently known unconditional bound of $\frac{2}{n} + O(n^{-2})$. We also improve an unconditional upper …
Let $Θ= (θ_1,θ_2,θ_3)\in \mathbb{R}^3$. Suppose that $1,θ_1,θ_2,θ_3$ are linearly independent over $\mathbb{Z}$. For Diophantine exponents $$ α(Θ) = \sup \{γ>0:\,\,\, \limsup_{t\to +\infty} t^γψ_Θ(t) <+\infty \} ,$$ $$β(Θ) = \sup \{γ>0:\,\,\, \liminf_{t\to +\infty} t^γψ_Θ(t) <+\infty\} $$ we prove $$ β(Θ) \ge {1/2} ({α(Θ)}/{1-α(Θ)} +\sqrt{{α(Θ)}/{1-α(Θ)})^2 +{4α(Θ)}/{1-α(Θ)}}) α(Θ) $$
We study the problem of best approximations of a vector $α\in{\mathbb R}^n$ by rational vectors of a lattice $Λ\subset {\mathbb R}^n$ whose common denominator is bounded. To this end we introduce successive minima for a periodic lattice structure and extend some classical results from geometry of numbers to this structure. This leads to bounds for the best approximation problem which generalize and improve former results.
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.
Simultaneous diophantine approximation for a restricted class of pairs of real numbers
该研究针对一类受限制的不可逼近数对$(α,β)$，通过利用系数依赖其丢番图性质的三次方程根的局部化方法及连分数收敛分母性质，证明了Littlewood猜想对这类数对成立。
There are abundant results on Diophantine approximation over fields of positive characteristic (see the survey papers [13, 25]), but there is very little information about simultaneous approximation. In this paper, we develop a technique of \geometry of numbers" in positive characteristic, so that we may generalize some of the classical results on simultaneous approximation to the case of function fields. More precisely, we approximate a finite set of Laurent series by rational functions with a common denominator. In particular, the lower bound results we obtain may be regarded as a high dimen…
The Hausdorff dimension of the set of simultaneously tau well approximable points lying on a curve defined by a polynomial P(X)+alpha, where P(X) is a polynomial with integer coefficients and alpha is in R, is studied when tau is larger than the degree of P(X). This provides the first results related to the computation of the Hausdorff dimension of the set of well approximable points lying on a curve which is not defined by a polynomial with integer coefficients. The proofs of the results also include the study of problems in Diophantine approximation in the case where the numerators and the d…
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。
We obtain some new inequalities between the ordinary and the uniform Diophantine exponents for simultaneous Diophantine approximation to four real numbers.
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.
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…
Simultaneous Diophantine approximation is concerned with the approximation of a point $\mathbf x\in\mathbb R^d$ by points $\mathbf r\in\mathbb Q^d$, with a view towards jointly minimizing the quantities $\|\mathbf x - \mathbf r\|$ and $H(\mathbf r)$. Here $H(\mathbf r)$ is the so-called "standard height" of the rational point $\mathbf r$. In this paper the authors ask: What changes if we replace the standard height function by a different one? As it turns out, this change leads to dramatic differences from the classical theory and requires the development of new methods. We discuss three examp…
We establish several new inequalities linking classical exponents of Diophantine approximation associated to a real vector $\underlineξ=(ξ,ξ^{2},\ldots,ξ^{N})$, in various dimensions $N$. We thereby obtain variants, and partly refinements, of recent results of Badziahin and Bugeaud. We further implicitly recover inequalities of Bugeaud and Laurent as special cases, with new proofs. Similar estimates concerning general real vectors (not on the Veronese curve) with $\mathbb{Q}$-linearly independent coordinates are addressed as well. Our method is based on Minkowski's Second Convex Body Theorem, …
Let Q be an infinite set of positive integers. Denote by W_{τ, n}(Q) (resp. W_{τ, n}) the set of points in dimension n simultaneously τ--approximable by infinitely many rationals with denominators in Q (resp. in N*). A non--trivial lower bound for the Hausdorff dimension of the liminf set W_{τ, n}\W_{τ, n}(Q) is established when n>1 and τ>1+1/(n-1) in the case where the set Q satisfies some divisibility properties. The computation of the actual value of this Hausdorff dimension as well as the one--dimensional analogue of the problem are also discussed.
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…
In 1998, Kleinbock & Margulis established a conjecture of V.G. Sprindzuk in metrical Diophantine approximation (and indeed the stronger Baker-Sprindzuk conjecture). In essence the conjecture stated that the simultaneous homogeneous Diophantine exponent $w_{0}(\vv x) = 1/n$ for almost every point $\vv x$ on a non-degenerate submanifold $\cM$ of $\R^n$. In this paper the simultaneous inhomogeneous analogue of Sprindzuk's conjecture is established. More precisely, for any `inhomogeneous' vector $\bmθ\in\R^n$ we prove that the simultaneous inhomogeneous Diophantine exponent $w_{0}(\vv x, \bmθ)= 1/…
Simultaneous Diophantine approximation on the three dimensional Veronese curve
本研究计算了三维Veronese曲线上特定参数范围内联立λ-良逼近点集的Hausdorff维数，验证了Beresnevich和Yang猜想的下界部分，使该曲线成为高维空间中首个满足该猜想下界的非退化曲线。
This paper develops the metric theory of simultaneous inhomogeneous Diophantine approximation on a planar curve with respect to multiple approximating functions. Our results naturally generalize the homogeneous Lebesgue measure and Hausdor? dimension results for the sets of simultaneously well-approximable points on planar curves, established in [2, 6, 10, 20].
By applying Schmidt's lattice method, we prove results on simultaneous Diophantine approximation modulo 1 for systems of polynomials in a single prime variable provided that certain local conditions are met.
We present a sharpening of nondivergence estimates for unipotent (or more generally polynomial-like) flows on homogeneous spaces. Applied to metric Diophantine approximation, it yields precise formulas for Diophantine exponents of affine subspaces of $\Bbb R^n$ and their nondegenerate submanifolds.
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。
Affine Coordinate Subspaces and Strong Khintchine Convergence
本文证明正维数正余维数的仿射坐标子空间不具强Khintchine收敛性，并结合Huang判据给出充分丢番图条件，使其具Khintchine型而非强Khintchine型。
An upper bound of the Hausdorff dimension of singular vectors on affine subspaces
本文研究了$\mathbb{R}^n$中仿射子空间上的奇异向量集合，并给出了其豪斯多夫维数的一个上界，该上界由仿射子空间参数的丢番图指数决定。
Core abaci and Diophantine equations I: fundamental weight
本文定义并研究了任意电荷下的经典仿射类型核心abaci，通过关联核心abaci与权重及仿射威利群元素，证明了β的高度等于原子长度，并解决了广义开放问题，同时建立了经典仿射类型的丢番图方程，并证明某些类别的解由核心abaci完全参数化。
S-arithmetic Khintchine-type theorem for products of non-degenerate analytic p-adic manifolds is proved for the convergence case. In the p-adic case the divergence part is also obtained.
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.
This brief survey deals with multi-dimensional Diophantine approximations in sense of linear form and with simultaneous Diophantine approximations. We discuss the phenomenon of degenerate dimension of linear subspaces generated by the best Diophantine approximations. Originally most of these results have been established by the author in few years ago. Here we collect all of them together and give some new formulations.
A New Diophantine Approximation Inequality on Surfaces and Its Applications
本文证明了曲面上闭子概形的丢番图逼近不等式，推广了Ru-Vojta和Heier-Levin的近期结果，并应用于仿射曲面上的整点退化、最大公因数不等式及S-单位方程，同时在复分析背景下建立了第二主定理型不等式。
In this paper we present a new method of solving certain quartic and higher degree homogeneous polynomial diophantine equations in four variables. The method can also be extended to solve simultaneous homogeneous polynomial diophantine equations, in five or more variables, with one of the equations being of degree $\geq 4$. We show that, under certain conditions, the method yields an arbitrarily large number of integer solutions of such diophantine equations and diophantine systems, some examples being a sextic equation in four variables, a tenth degree equation in six variables, and two simul…
For $K$ a cubic field with only one real embedding and $α,β\in K$, we show how to construct an increasing sequence $\{m_n\}$ of positive integers and a subsequence $\{ψ_n\}$ such that (for some constructible constants $C_1,C_2>0$) $\max\{\|m_nα\|,\|m_nβ\|\}<\frac{C_1}{m_n^{1/2}}$ and $\|ψ_nα\|<\frac{C_2}{ψ_n^{1/2}\log ψ_n}$ for all $n$. As a consequence, we have $ψ_n\|ψ_nα\|\|ψ_nβ\|<\frac{C_1 C_2}{\log ψ_n}$, thus giving an effective proof of Littlewood's conjecture for the pair $(α,β)$. Our proofs are elementary and use only standard results from algebraic number theory and the theory of cont…
In recent years, the ergodic theory of group actions on homogeneous spaces has played a significant role in the metric theory of Diophantine approximation. We survey some recent developments with special emphasis on Diophantine properties of affine subspaces and their submanifolds.
We show that affine coordinate subspaces of dimension at least two in Euclidean space are of Khintchine type for divergence. For affine coordinate subspaces of dimension one, we prove a result which depends on the dual Diophantine type of the basepoint of the subspace. These results provide evidence for the conjecture that all affine subspaces of Euclidean space are of Khintchine type for divergence.
Diophantine equations in primes: density of prime points on affine hypersurfaces II
本文证明在满足局部条件且维数足够大的仿射超曲面上，齐次丢番图方程存在素数坐标整数解，通过优化 Heath-Brown 恒等式方法改进了先前基于 Vaughan 恒等式的维数条件。
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.
In this paper, we introduce an algebro-geometric formulation for Siegel's theorem using an improvement of Lang's version of Roth's theorem over finitely generated fields of characteristic zero. In fact, we prove that, for an affine open curve in an irreducible smooth curve of genus at least one, any finitely generated subgroup of the additive group of the affine ambient space intersects the open curve in only finitely many points. This was proved only for finitely generated subgroups defined over a localization of the ring of integers of a number field by Mahler and others.
We solve the convergence case of the generalized Baker-Schmidt problem for simultaneous approximation on affine subspaces, under natural diophantine type conditions. In one of our theorems, we do not require monotonicity on the approximation function. In order to prove these results, we establish asymptotic formulae for the number of rational points close to an affine subspace. One key ingredient is a sharp upper bound on a certain sum of reciprocals of fractional parts associated with the matrix defining the affine subspace.
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.
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