Cor Kraaikamp, Ionica Smeets
In 2004, J.C. Tong found bounds for the approximation quality of a regular continued fraction convergent of a rational number, expressed in bounds for both the previous and next approximation. We sharpen his results with a geometric method and give both sharp upper and lower bounds. We also calculate the asymptotic frequency that these bounds occur.
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The present paper establishes qunatitative estimates on the rate of diophantine approximation in homogeneous varieties of semisimple algebraic groups. The estimates established generalize and improve previous ones, and are sharp in a number of cases. We show that the rate of diophantine approximation is controlled by the spectrum of the automorphic representation, and is thus subject to the generalised Ramanujan conjectures.
We refine Khintchine Transference Principle which relates the measure of simultaneous rational approximation of an $n$ real numbers with the measure of linear independence of these $n$ numbers. Khintchine's inequalities are known to be optimal. However, they may be sharpened by taking into account two further uniform exponents.
Let $Θ$ be a point in ${\bf R}^n$. We split the classical Khintchine's Transference Principle into $n-1$ intermediate estimates which connect exponents $ω_d(Θ)$ measuring the sharpness of the approximation to $Θ$ by linear rational varieties of dimension $d$, for $0\le d \le n-1$. We also review old and recent results related to these $n$ exponents.
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。
Sharpening Vahlen's result in Diophantine approximation
本文通过几何方法为丢番图逼近中的逼近系数提供更锐利的界，改进了Vahlen 1895年的结果，并为改进Borel经典常数提供了新见解。
Sharp Transitions for Localized Solutions to a Diophantine Inequality
针对近对角范围变量的丢番图不等式解的存在性，证明了解的个数随参数c呈现 sharp 相变，c>ω时大R必存在解，c<ω时大R可无解，结果类似华林问题的对应结论且可能更强。
We prove a sharp analogue of Minkowski's inhomogeneous approximation theorem over fields of power series $\mathbb{F}_q((T^{-1}))$. Furthermore, we study the approximation to a given point $\underline{y}$ in $\mathbb{F}_q((T^{-1}))^2$ by the $SL_2(\mathbb{F}_q[T])$-orbit of a given point $\underline{x}$ in $\mathbb{F}_q((T^{-1}))^2$.
This paper provides asymptotics with a sharp error term for the Dirichlet summatory function of a certain class of arithmetic functions. The result applies, e.g., to the sums over r^2(n) and r(n^3), where r(m) denotes the number of ways to write m as a sum of two squares of integers.
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. …
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 …
We solve the problem of giving sharp asymptotic bounds on the Hausdorff dimensions of certain sets of badly approximable matrices, thus improving results of Broderick and Kleinbock (preprint 2013) as well as Weil (preprint 2013), and generalizing to higher dimensions those of Kurzweil ('51) and Hensley ('92). In addition we use our technique to compute the Hausdorff $f$-measure of the set of matrices which are not $ψ$-approximable, given a dimension function $f$ and a function $ψ:(0,\infty)\to (0,\infty)$. This complements earlier work by Dickinson and Velani ('97) who found the Hausdorff $f$-…
On the Diophantine equation $X^2+p^m=p_1^{n_1}p_2^{n_2}$
该研究针对含三个不同奇素数的丢番图方程 $X^2+p^m=p_1^{n_1}p_2^{n_2}$，通过二次域因式分解等方法得到相关指数的显式界，证明固定素数时方程解的个数有限。
Sharpening Borel's result in Diophantine approximation
本文通过为连续三个逼近系数最小值提供更精确界限，锐化了博雷尔1903年在丢番图逼近中的结果，聚焦于\(a_{n + 1}=1\)的情况进行研究。
Diophantine approximation in metric space
本文提出在任意全有界度量空间中用抽象有理数替代有理数研究丢番图逼近，证明Jarnik-Besicovitch型维数界并探讨其最优性。
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。
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…
Diophantine Approximation in local function fields via Bruhat-Tit trees
基于 Bruhat-Tits 树的算术商理论，获得了全局函数域上丢番图逼近的 Dirichlet 风格定理，并计算了相关常数与逼近集合测度。
Recently in joint work with E. Sert, we proved sharp boundedness results on discrete fractional integral operators along binary quadratic forms. Present work vastly enhances the scope of those results by extending boundedness to bivariate quadratic polynomials. We achieve this in part by establishing connections to problems on concentration of lattice points on short arcs of conics, whence we study discrete fractional integrals and lattice point concentration from a unified perspective via tools of sieving and diophantine approximation, and prove theorems that are of interest to researchers in…
We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces. This approach yields a new proof of a conjecture of Mahler, originally settled by V. Sprindzhuk in 1964. We also prove several related hypotheses of A. Baker and V. Sprindzhuk formulated in 1970s. The core of the proof is a theorem which generalizes and sharpens earlier results on non-divergence of unipotent flows on the space of lattices.
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.
We generalise M. M. Skriganov's notion of weak admissibility for lattices to include standard lattices occurring in Diophantine approximation and algebraic number theory, and we prove estimates for the number of lattice points in sets such as aligned boxes. Our result improves on Skriganov's celebrated counting result if the box is sufficiently distorted, the lattice is not admissible, and, e.g., symplectic or orthogonal. We establish a criterion under which our error term is sharp, and we provide examples in dimensions $2$ and $3$ using continued fractions. We also establish a similar countin…
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。
Let ${n_k}$ be an increasing lacunary sequence, i.e., $n_{k+1}/n_k>1+r$ for some $r>0$. In 1987, P. Erdos asked for the chromatic number of a graph $G$ on the integers, where two integers $a,b$ are connected by an edge iff their difference $|a-b|$ is in the sequence ${n_k}$. Y. Katznelson found a connection to a Diophantine approximation problem (also due to Erdos): the existence of $x$ in $(0,1)$ such that all the multiples $n_j x$ are at least distance $δ(x)>0$ from the set of integers. Katznelson bounded the chromatic number of $G$ by $Cr^{-2}|\log r|$. We apply the Lovász local lemma to es…
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。
Given $n\in\mathbb{N}$ and $τ>\frac1n$, let $\mathcal{S}_n(τ)$ denote the classical set of $τ$-approximable points in $\mathbb{R}^n$, which consists of ${\bf x}\in \mathbb{R}^n$ that lie within distance $q^{-τ-1}$ from the lattice $\frac1q\mathbb{Z}^n$ for infinitely many $q\in\mathbb{N}$. In pioneering work, Kleinbock $\&$ Margulis showed that for any non-degenerate submanifold $\mathcal{M}$ of $\mathbb{R}^n$ and any $τ>\frac1n$ almost all points on $\mathcal{M}$ are not $τ$-approximable. Numerous subsequent papers have been geared towards strengthening this result through investigating the H…
We study the general problem of extremality for metric Diophantine approximation on submanifolds of matrices. We formulate a criterion for extremality in terms of a certain family of algebraic obstructions and show that it is sharp. In general, the almost sure diophantine exponent of a submanifold is shown to depend only on its Zariski closure, and when the latter is defined over the rational numbers, we prove that the exponent is rational and give a method to effectively compute it. This method is applied to a number of cases of interest, in particular, we manage to determine the diophantine …
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。
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。
Dimension of Diophantine approximation and some applications in harmonic analysis
本文通过构造基于Diophantine逼近的新集合，探讨了其在调和分析中的应用，包括Hausdorff维数、傅里叶维数以及测度的精确性问题。
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。
On connectedness in the parametric geometry of numbers
本文利用多线性代数与铅笔簇建立数的参数几何中连通性的判据，并据此改进Schmidt和Summerer关于丢番图逼近的连通性结果。
We prove new parameterization theorems for sets definable in the structure $\mathbb{R}_{an}$ (i.e. for globally subanalytic sets) which are uniform for definable families of such sets. We treat both $C^r$-parameterization and (mild) analytic parameterization. In the former case we establish a polynomial (in $r$) bound (depending only on the given family) for the number of parameterizing functions. However, since uniformity is impossible in the latter case (as was shown by Yomdin via a very simple family of algebraic sets), we introduce a new notion, analytic quasi-parameterization (where many-…
Sharp density discrepancy for cut and project sets: An approach via lattice point counting
本文研究切割与投影集的点计数渐近波动，获得了依赖于格丢番图性质的偏差一致上下界，并证明了其尖锐性，同时找到了与格双Lipschitz等价的最大切割与投影集类。
For $ E\subset \mathbb{F}_q^d$, let $Δ(E)$ denote the distance set determined by pairs of points in $E$. By using additive energies of sets on a paraboloid, Koh, Pham, Shen, and Vinh (2020) proved that if $E,F\subset \mathbb{F}_q^d $ are subsets with $|E||F|\gg q^{d+\frac{1}{3}}$ then $|Δ(E)+Δ(F)|> q/2$. They also proved that the threshold $q^{d+\frac{1}{3}}$ is sharp when $|E|=|F|$. In this paper, we provide an improvement of this result in the unbalanced case, which is essentially sharp in odd dimensions. The most important tool in our proofs is an optimal $L^2$ restriction theorem for the s…
A sharp threshold for arithmetic effects on the tail probabilities of lacunary sums
本文研究间隙和尾部概率的算术效应，证明存在尖锐阈值：当丢番图方程解数满足$L(N,a,b,c) \le N / g_N$时，间隙和尾部概率在$t$≤$\sqrt{2 \log g_N}$时呈高斯行为，且该准则最优。
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。
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