Anish Ghosh
We prove that the convergence Khintchine theorem holds for affine hyperplanes whose parametrizing matrices satisfy a mild Diophantine condition. We use the dynamical method of Kleinbock-Margulis.
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In this paper we establish the convergence case of Khintchine's theorem for affine hyperplanes in function field of positive characteristic. Along with that, we also prove a quantitative version of the same. The main technique used in the proof is a dynamical result called `Quantitative nondivergence' due to D. Y. Kleinbock and G. A. Margulis [17].
There has been great interest in developing a theory of "Khintchine types" for manifolds embedded in Euclidean space, and considerable progress has been made for curved manifolds. We treat the case of translates of coordinate hyperplanes, decidedly flat manifolds. In our main results, we fix the value of one coordinate in Euclidean space and describe the set of points in the fiber over that fixed coordinate that are rationally approximable at a given rate. We identify translated coordinate hyperplanes for which there is a dichotomy as in Khintchine's Theorem: the set of rationally approximable…
In this article we prove a convergence S-arithmetic Khintchine-type theorem for product of non-degenerate v-adic manifolds, where one of them is the Archimedian place.
An analogue of the convergence part of the Khintchine-Groshev theorem, as well as its multiplicative version, is proved for nondegenerate smooth submanifolds in $\mathbb{R}^n$. The proof combines methods from metric number theory with a new approach involving the geometry of lattices in Euclidean spaces.
Rational points near manifolds and Khintchine theorem
本文为 $\mathbb{R}^n$ 中任意非退化流形建立了 Khintchine 型定理，移除了先前的解析性假设，并基于定量非发散结果和对偶论证给出了 Jarník 型精化及非齐次设定下的结果。
Simultaneous Khintchine theorem on manifolds in positive characteristics: convergence case
本文在正特征局部域上，针对解析非平面流形证明了辛钦定理的收敛情况，通过扩展函数域处理非单调近似函数，并获得新的数论几何结果。
Generalized Lévy-Khintchine Theorems and a Conjecture of Y. Cheung
本研究解决了Y. Cheung关于任意范数下Lévy-Khintchine型定理的猜想，建立了高维Doeblin-Lenstra定律类似物，还将相关结果推广到三分康托集等多种分形与曲线支撑的测度上。
In this paper we study random iterated function systems. Our main result gives sufficient conditions for an analogue of a well known theorem due to Khintchine from Diophantine approximation to hold almost surely for stochastically self-similar and self-affine random iterated function systems.
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.
The main objective of this paper is to prove a Khintchine type theorem for divergence for linear Diophantine approximation on non-degenerate manifolds, which completes earlier results for convergence.
We show that affine subspaces of Euclidean space are of Khintchine type for divergence under certain multiplicative Diophantine conditions on the parametrizing matrix. This provides evidence towards the conjecture that all affine subspaces of Euclidean space are of Khintchine type for divergence, or that Khintchine's theorem still holds when restricted to the subspace. This result is proved as a special case of a more general Hausdorff measure result from which the Hausdorff dimension of W(τ) intersected with an appropriate subspace is also obtained.
Khintchine-type theorems for weighted uniform inhomogeneous approximations via transference principle
该研究推广了Kleinbock与Wadleigh提出的一致非齐次丢番图逼近零一律，引入任意权函数，基于转移原理给出简洁证明，刻画了带固定矩阵的$g$-Dirichlet对集合与坏逼近矩阵的性质，且相关测度结论适用于加权丢番图逼近。
In his 1960 paper, Schmidt studied a quantitative type of Khintchine-Groshev theorem for general (higher) dimensions. Recently, a new proof of the theorem was found, which made it possible to relax the dimensional constraint and more generally, to add on the congruence condition by M. Alam, A. Ghosh, and S. Yu. In this paper, we generalize this new approach to S-arithmetic spaces and obtain a quantitative version of an S-arithmetic Khintchine-Groshev theorem. In fact, we consider a new S-arithmetic analog of Diophantine approximation, which is different from the one formerly established (see t…
Lévy-Khintchine Theorems: effective results and central limit theorems
本文提出基于齐性动力学的新方法以量化 Lévy-Khintchine 定理，将有效结果推广到高维联立丢番图逼近，并证明所有维度最佳逼近的中心极限定理。
This work addresses problems on simultaneous Diophantine approximation on fractals, motivated by a long standing problem of Mahler regarding Cantor's middle $1/3$ set. We obtain the first instances where a complete analogue of Khintchine's Theorem holds for fractal measures. Our results apply to fractals which are self-similar by a system of rational similarities of $\mathbb{R}^d$ (for any $d\geq 1$) and have sufficiently small Hausdorff co-dimension. A concrete example of such measures in the context of Mahler's problem is the Hausdorff measure on the "middle $1/5$ Cantor set"; i.e. the set o…
We discuss Khintchine's theorem on regular matrices (1948) and its exposition in Cassels' book (1957). The paper is written in Russian.
In the paper we provide measure estimates for the set of numbers whose sequence of products of continued fraction partial quotients $M_n = a_1 \ldots a_n$ has exponential growth with rate close to the one predicted by Khintchine's theorem, i.e. for which \begin{equation*} e^{(κ- T)n} \leqslant M_n \leqslant e^{(κ+ T)n} \end{equation*} for a fixed $T > 0$ and all $n$ greater than some fixed integer $N$, where $e^κ= 2.685\ldots$ is the Khintchine constant. Choosing $N$ large enough the measure can be made arbitrarily close to full, for any given $T$. The bounds are not of asymptotic nature, but …
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…
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。
Khintchine's theorem for inhomogeneous simultaneous approximation with polynomial decay
本文研究非齐次同时逼近中多项式衰减条件下的辛钦定理，解决(n,m)=(1,2)情况下的单调性条件去除问题。
Cone upgrade for effective equidistribution and weighted Khintchine--Schmidt theorem on fractals
该研究针对实线上具有共同压缩比的自相似测度乘积，证明了加权Khintchine-Schmidt定理的分形类似结果，通过提出锥提升方法，得到Khalil等人定理的部分有效形式。
Inhomogeneous Khintchine-Groshev theorem without monotonicity
本文证明了Allen和Ramírez关于非齐次Khintchine-Groshev定理无单调性条件的猜想在(n,m)=(2,1)及有理参数情形下的(n,m)=(1,2)成立。
Under the assumption that the approximating function $ψ$ is monotonic, the classical Khintchine-Groshev theorem provides an elegant probabilistic criterion for the Lebesgue measure of the set of $ψ$-approximable matrices in $\R^{mn}$. The famous Duffin-Schaeffer counterexample shows that the monotonicity assumption on $ψ$ is absolutely necessary when $m=n=1$. On the other hand, it is known that monotonicity is not necessary when $n > 2$ (Sprindzuk) or when $n=1$ and $m>1$ (Gallagher). Surprisingly, when $n=2$ the situation is unresolved. We deal with this remaining case and thereby remove all …
The inhomogeneous Khintchine Theorem in dimension two
本文证明了二维情况下无需单调性假设的非均匀Khintchine定理成立，解决了非均匀Diophantine逼近度量理论中最后一个未解决的案例，并确认了一个关于线性形式系统逼近的Khintchine-Groshev型定理的猜想。
The Duffin--Schaeffer Conjecture answers a question on how well one can approximate irrationals by rational numbers in reduced form (an imposed condition) where the accuracy of the approximation depends on the rational number. It can be viewed as an analogue to Khintchine's Theorem with the added restriction of only allowing rationals in reduced form. Other conditions such as numerator or denominator a prime, a square-free integer, or an element of a particular arithmetic progression, etc. have also been imposed and analogues of Khintchine's Theorem studied. We prove versions of Khintchine's T…
An asymptotic formula which holds almost everywhere is obtained for the number of solutions to the Diophantine inequalities |qA-p|<ψ(|q|), where A is an n by m matrix (m>1) over the field of formal Laurent series with coefficients from a finite field, and p and q are vectors of polynomials over the same finite field.
Khintchine's theorem is a classical result from metric number theory which relates the Lebesgue measure of certain limsup sets with the convergence/divergence of naturally occurring volume sums. In this paper we ask whether an analogous result holds for iterated function systems (IFSs). We say that an IFS is approximation regular if we observe Khintchine type behaviour, i.e., if the size of certain limsup sets defined using the IFS is determined by the convergence/divergence of naturally occurring sums. We prove that an IFS is approximation regular if it consists of conformal mappings and sati…
Extremal affine subspaces and Khintchine-Jarník type theorems
本文证明Kleinbock猜想，完整分类极值仿射子空间，并给出Khintchine型及Jarník型定理的分类，基于有理点计数新估计，运用多维大筛不等式。
Our main result concerns a perturbation of a classic theorem of Khintchine in Diophantine approximation. We give sufficient conditions on a sequence of positive real numbers $(ψ_n)_{n \in \mathbb{N}}$ and differentiable functions $(φ_n: J \to \mathbb{R})_{n \in \mathbb{N}}$ so that for Lebesgue-a.e. $θ\in J$, the inequality $\| nθ+ φ_n(θ) \| \leq ψ_n$ has infinitely many solutions. The main novelty is that the magnitude of the perturbation $|φ_n(θ)|$ is allowed to exceed $ψ_n$, changing the usual "shrinking targets" problem into a "shifting targets" problem. As an application of the main resul…
The classical Khintchine-Groshev theorem is a generalization of Khintchine's theorem on simultaneous Diophantine approximation, from approximation of points in $\mathbb R^m$ to approximation of systems of linear forms in $\mathbb R^{nm}$. In this paper, we present an inhomogeneous version of the Khintchine-Groshev theorem which does not carry a monotonicity assumption when $nm>2$. Our results bring the inhomogeneous theory almost in line with the homogeneous theory, where it is known by a result of Beresnevich and Velani (2010) that monotonicity is not required when $nm>1$. That result resolve…
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.
Toward Khintchine's theorem with a moving target: extra divergence or finitely centered target
本文证明Szüsz非齐次Khintchine定理在移动目标（参数γ随分母变化）下，在额外发散或有限中心条件下仍成立，并得到有限着色版本。
We prove versions of Khintchine's Theorem (1924) for approximations by rational numbers whose numerators lie in randomly chosen sets of integers, and we explore the extent to which the monotonicity assumption can be removed. Roughly speaking, we show that if the number of available fractions for each denominator grows too fast, then the monotonicity assumption cannot be removed. There are questions in this random setting which may be seen as cognates of the Duffin-Schaeffer Conjecture (1941), and are likely to be more accessible. We point out that the direct random analogue of the Duffin-Schae…
New results towards the Duffin-Schaeffer conjecture, which is a fundamental unsolved problem in metric number theory, have been established recently assuming extra divergence. Given a non-negative function $ψ: \mathbb{N}\to\mathbb{R}$ we denote by $W(ψ)$ the set of all $x\in\mathbb{R}$ such that $|nx-a|<ψ(n)$ for infinitely many $a,n$. Analogously, denote $W'(ψ)$ if we additionally require $a,n$ to be coprime. Aistleitner et al. [1] proved that $W'(ψ)$ is of full Lebesgue measure if there exist an $\varepsilon>0$ such that $\sum_{n=2}^\inftyψ(n)φ(n)/(n(\log n)^\varepsilon)=\infty$. This result…
This work has been motivated by recent papers that quantify the density of values of generic quadratic forms and other polynomials at integer points, in particular ones that use Rogers' second moment estimates. In this paper we establish such results in a very general framework. Given any subhomogeneous function (a notion to be defined) $f: \mathbb{R}^n \to \mathbb{R}$, we derive a necessary and sufficient condition on the approximating function $ψ$ for guaranteeing that a generic element $f\circ g$ in the $G$-orbit of $f$ is $ψ$-approximable; that is, $|f\circ g(\mathbf{v})| \le ψ(\|\mathbf{v…
Khintchine dichotomy for self-similar measures
本文研究了自相似测度上的辛钦定理，通过分析 SL_{2}(R)/SL_{2}(Z) 上的随机游走等分布性，解决了中 thirds 卡ntor 集的豪斯多夫测度问题。
Quantitative Khintchine on the parabola with non-monotonic approximation functions
该研究针对抛物线上的点，证明了度量丢番图逼近中Khintchine定理收敛情形的定量版本，核心是突破了逼近函数需单调的限制，还用到了Burgess界等经典数论结果。
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。
Recently, Ghosh \& Haynes \cite{HG} proved a Khintchine-type result for the problem of Diophantine approximation in certain projective spaces. In this note we complement their result by observing that a Jarn\'ık-type result also holds for `badly approximable' points in real projective space. In particular, we prove that the natural analogue in projective space of the classical set of badly approximable numbers has full Hausdorff dimension when intersected with certain compact subsets of real projective space. Furthermore, we also establish an analogue of Khintchine's theorem for convergence re…
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.
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…
We prove a quantitative theorem for Diophantine approximation by rational points on spheres. Our results are valid for arbitrary unimodular lattices and we further prove 'spiraling' results for the direction of approximates. These results are quantitative generalizations of the Khintchine-type theorem on spheres proved by Kleinbock and Merrill.
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 paper the absolute value or distance from the origin analogue of the classical Khintchine-Groshev theorem is established for a single linear form with a `slowly decreasing' error function.
Khintchine's and Dyson's transference theorems can be very easily deduced from Mahler's transference theorem. In the multiplicative setting an obstacle appears, which does not allow deducing the multiplicative transference theorem immediately from Mahler's theorem. Some extra considerations are required, for instance, induction by the dimension. In this paper we propose an analogue of Mahler's theorem which implies the multiplicative transference theorem immediately.
We refine metrical statements in the style of the Khintchine-Groshev Theorem by requiring certain coprimality constraints on the coordinates of the integer solutions.
A complete p-adic Khintchine type theorem for approximation by p-adic algebraic numbers is established.
We outline a proof of an analogue of Khintchine's Theorem in R^2, where the ordinary height is replaced by a distance function satisfying an irrationality condition as well as certain decay and symmetry conditions.
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…
Quantitative inhomogeneous Diophantine approximation for systems of linear forms
本文在非齐次参数为非Liouville无理数时，证明了$(m,n)=(1,2)$情形下非齐次Khintchine-Groshev定理无需单调性假设，并进一步得到了具有几乎平方根抵消的渐近公式。
In this paper we establish a general form of the Mass Transference Principle for systems of linear forms conjectured in [1]. We also present a number of applications of this result to problems in Diophantine approximation. These include a general transference of Lebesgue measure Khintchine-Groshev type theorems to Hausdorff measure statements. The statements we obtain are applicable in both the homogeneous and inhomogeneous settings as well as allowing transference under any additional constraints on approximating integer points. In particular, we establish Hausdorff measure counterparts of so…
Analogues of the classical theorems of Khintchine, Jarnik and Jarnik-Besicovitch in the metrical theory of Diophantine approximation are established for quaternions by applying results on the measure of general `lim sup' sets.
In this short paper we prove a quantitative version of the Khintchine-Groshev Theorem with congruence conditions. Our argument relies on a classical argument of Schmidt on counting generic lattice points, which in turn relies on a certain variance bound on the space of lattices.
This paper is motivated by recent applications of Diophantine approximation in electronics, in particular, in the rapidly developing area of Interference Alignment. Some remarkable advances in this area give substantial credit to the fundamental Khintchine-Groshev Theorem and, in particular, to its far reaching generalisation for submanifolds of a Euclidean space. With a view towards the aforementioned applications, here we introduce and prove quantitative explicit generalisations of the Khintchine-Groshev Theorem for non-degenerate submanifolds of $\mathbb{R}^n$. The importance of such quanti…
We study approximation in the unit interval by rational numbers whose numerators are selected randomly with certain probabilities. Previous work showed that an analogue of Khintchine's Theorem holds in a similar random model and raised the question of when the monotonicity assumption can be removed. Informally speaking, we show that if the probabilities in our model decay sufficiently fast as the denominator increases, then a Khintchine-like statement holds without a monotonicity assumption. Although our rate of decay of probabilities is unlikely to be optimal, it is known that such a result w…
We prove a weighted analogue of the Khintchine-Groshev Theorem, where the distance to the nearest integer is replaced by the absolute value. This is subsequently applied to proving the optimality of several linear independence criteria over the field of rational numbers.
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.
We prove a Khintchine type theorem for approximation of elements in the Cantor set, as a subset of the formal Laurent series over $\mathbb{F}_3$, by rational functions of a specific type. Furthermore we construct elements in the Cantor set with any prescribed irrationality exponent $\geq 2$.
We introduce a novel approach to Bertini irreducibility theorems over an arbitrary field, based on random hyperplane slicing over a finite field. Extending a result of Benoist, we prove that for a morphism $ϕ\colon X \to \mathbb{P}^n$ such that $X$ is geometrically irreducible and the nonempty fibers of $ϕ$ all have the same dimension, the locus of hyperplanes $H$ such that $ϕ^{-1} H$ is not geometrically irreducible has dimension at most $\operatorname{codim} ϕ(X)+1$. We give an application to monodromy groups above hyperplane sections.
We first study hyperplane sections of some singular schemes over a field. We prove a Bertini theorem for the log smoothness of generic hyperplane sections of a large class of log smooth schemes over a log point. We also give an abstract generalization of the usual Bertini theorem for smoothness. We then apply this result to the study of hyperplane sections of regular schemes over a complete discrete valuation ring of characteristic zero with algebraically closed residue field. Lastly, give an exposition of a result of Faltings which states that one can compute the $p$-adic nearby cycles of smo…
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