We show that for any $ε<1$ and any $\mathcal{T}$ `drifting away from walls', Dirichlet's Theorem cannot be $ε$-improved along $\mathcal{T}$ for Lebesgue almost every system of linear forms $Y$ (see the paper for definitions). In the case $m = 1$ we also show that for a large class of measures $μ$ there is $ε_0>0$ such that for any drifting away from walls $\mathcal{T}$, any $ε<ε_0$, and for $μ$-almost every $Y$, Dirichlet's Theorem cannot be $ε$-improved along $\mathcal{T}$. These measures include natural measures on sufficiently regular smooth manifolds and fractals.
Our results extend those of several authors beginning with the work of Davenport and Schmidt done in late 1960s. The proofs rely on a translation of the problem into a dynamical one regarding the action of a diagonal semigroup on the space $\text{SL}_{m+n}(\mathbb{R})/\text{SL}_{m+n}(\mathbb{Z})$.
We study metric Diophantine approximation in local fields of positive characteristic. Specifically, we study the problem of improving Dirichlet's theorem in Diophantine approximation and prove very general results in this context.
This note pushes further the discussion about relations between Dirichlet improvable, badly approximable and singular points held in recent joint work with Beresnevich, Guan, Velani and Ramirez, by considering Diophantine sets extending the notion of badly approximability.
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 consider improvements of Dirichlet's Theorem on space of matrices $M_{m,n}(R)$. It is shown that for a certain class of fractals $K\subset [0,1]^{mn}\subset M_{m,n}(R)$ of local maximal dimension Dirichlet's Theorem cannot be improved almost everywhere. This is shown using entropy and dynamics on homogeneous spaces of Lie groups.
The main goal of this work is to establish quantitative nondivergence estimates for flows on homogeneous spaces of products of real and $p$-adic Lie groups. These results have applications both to ergodic theory and to Diophantine approximation. Namely, earlier results of Dani (finiteness of locally finite ergodic unipotent-invariant measures on real homogeneous spaces) and Kleinbock-Margulis (strong extremality of nondegenerate submanifolds of $\Bbb R^n$) are generalized to the $S$-arithmetic setting.
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…
Let $f$ be a homogeneous polynomial with rational coefficients in $d$ variables. We prove several results concerning uniform simultaneous approximation to points on the graph of $f$, as well as on the hypersurface $\{f(x_1,\dots,x_d) = 1\}$. The results are first stated for the case $f(x_1,\dots,x_d) = x_1^2+\dots+x_d^2,$ which is of particular interest.
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.
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 $Y_0$ be a not very well approximable $m\times n$ matrix, and let $M$ be a connected analytic submanifold in the space of $m\times n$ matrices containing $Y_0$. Then almost all $Y\in M$ are not very well approximable. This and other similar statements are cast in terms of properties of certain orbits on homogeneous spaces and deduced from quantitative nondivergence estimates for `quasi-polynomial' flows on on the space of lattices.
In this article, we study an analytic curve $φ: I=[a,b]\rightarrow \mathrm{M}(n\times n, \mathbb{R})$ in the space of $n$ by $n$ real matrices, and show that if $φ$ satisfies certain geometric conditions, then for almost every point on the curve, the Diophantine approximation given by Dirichlet's Theorem is not improvable. To do this, we embed the curve into some homogeneous space $G/Γ$, and prove that under the action of some expanding diagonal flow $A= \{a(t): t \in \mathbb{R}\}$, the expanding curves tend to be equidistributed in $G/Γ$, as $t \rightarrow +\infty$. This solves a special case…
We show that for almost all points on any analytic curve on R^{k} which is not contained in a proper affine subspace, the Dirichlet's theorem on simultaneous approximation, as well as its dual result for simultaneous approximation of linear forms, cannot be improved. The result is obtained by proving asymptotic equidistribution of evolution of a curve on a strongly unstable leaf under certain partially hyperbolic flow on the space of unimodular lattices in R^{k+1}. The proof involves ergodic properties of unipotent flows on homogeneous spaces.
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 …
Author developed a method in the paper, which, unlike the circle method of Hardy and Littlewood (CM), allows you to perform a lower estimate for the number of natural (integer) solutions of algebraic Diophantine equation with integer coefficients. It was found the lower estimate of the number of natural solutions to various types of homogeneous algebraic Diophantine equations with integer coefficients diagonal form with any number of variables using this method. Author obtained upper bound of the number of the natural solutions (using CM) of one type of homogeneous Diophantine equation for val…
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…
In Diophantine approximation, inhomogeneous problems are linked with homogeneous ones by means of the so-called Transference Theorems. We revisit this classical topic by introducing new exponents of Diophantine approximation. We prove that the exponent of approximation to a generic point in R^n by a system of n linear forms is equal to the inverse of the uniform homogeneous exponent associated to the system of dual forms.
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 …
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 extend the theory of simultaneous Diophantine approximation to infinite dimensions. Moreover, we discuss Dirichlet-type theorems in a very general framework and define what it means for such a theorem to be optimal. We show that optimality is implied by but does not imply the existence of badly approximable points.
We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space $G/Γ$ of a semisimple algebraic group $G$. We define two families of algebraic subvarieties of the associated partial flag variety $G/P$, which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of $m\times n$ real matrices whose image is not contained in any subvariety coming from these two families, the Dirichlet's theorem on simultaneous Dio…
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
We prove over fields of power series the analogues of several Diophantine approximation results obtained over the field of real numbers. In particular we establish the power series analogue of Kronecker's theorem for matrices, together with a quantitative form of it, which can also be seen as a transference inequality between uniform approximation and inhomogeneous approximation. Special attention is devoted to the one dimensional case. Namely, we give a necessary and sufficient condition on an irrational power series $α$ which ensures that, for some positive $\eps$, the set $$ \liminf_{Q \in …
small In this paper, we define $q$-analogues of Dirichlet's beta function at positive integers, which can be written as $β_q(s)=\sum_{k\geq1}\sum_{d|k}χ(k/d)d^{s-1}q^k$ for $s\in\N^*$, where $q$ is a complex number such that $|q|<1$ and $χ$ is the non trivial Dirichlet character modulo 4. For odd $s$, these expressions are connected with the automorphic world, in particular with Eisenstein series of level 4. From this, we derive through Nesterenko's work the transcendance of the numbers $β_q(2s+1)$ for $q$ algebraic such that $0<|q|<1$. Our main result concerns the nature of the numbers $β_q(2…
We apply nondivergence estimates for flows on homogeneous spaces to compute Diophantine exponents of affine subspaces of $\R^n$ and their nondegenerate submanifolds.
In a landmark paper, D.Y. Kleinbock and G.A. Margulis established the fundamental Baker-Sprindzuk conjecture on homogeneous Diophantine approximation on manifolds. Subsequently, there has been dramatic progress in this area of research. However, the techniques developed to date do not seem to be applicable to inhomogeneous approximation. Consequently, the theory of inhomogeneous Diophantine approximation on manifolds remains essentially non-existent. In this paper we develop an approach that enables us to transfer homogeneous statements to inhomogeneous ones. This is rather surprising as the i…
We find a parametric solution of an arbitrary symmetric homogeneous diophantine equation of 5th degree in 6 variables using two primitive solutions. We then generalize this approach to symmetric forms of any odd degree by proving the following results. (1) Every symmetric form of odd degree $n\ge 5$ in $6 \cdot 2^{n-5}$ variables has a rational parametric solution depending on $2n-8$ parameters. (2) Let $F(x_1, ..., x_N)$ be a symmetric form of odd degree $n\ge 5$ in $N=6 \cdot 2^{n-4}$ variables, and let $q$ be any rational number. Then the equation $F(x_i)=q$ has a rational parametric soluti…
In this paper we prove complete $p$-adic analogues of Kleinbock's theorems \cite{Kleinbock-extremal, Kleinbock-exponent} on inheritance of Diophantine exponents for affine subspaces. In particular, we answer in the affirmative (and in a stronger form), a conjecture of Kleinbock and Tomanov \cite{KT}, as well as a question of Kleinbock \cite{Kleinbock-exponent}. Our main innovation is the introduction of a new $p$-adic Diophantine exponent which is better suited to homogeneous dynamics, and which we show to be closely related to the exponent considered by Kleinbock and Tomanov.
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.
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…
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…
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…
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.
In this paper, we study an analytic curve $φ: I=[a,b]\rightarrow \mathrm{M}(m\times n, \mathbb{R})$ in the space of $m$ by $n$ real matrices, and show that if $φ$ satisfies certain geometric condition, then for almost every point on the curve, the Diophantine approximation given by Dirichlet's Theorem can not be improved. To do this, we embed the curve into some homogeneous space $G/Γ$, and prove that under the action of some expanding diagonal subgroup $A= \{a(t): t \in \mathbb{R}\}$, the translates of the curve tend to be equidistributed in $G/Γ$, as $t \rightarrow +\infty$.
We investigate the question of how well points on a nondegenerate $k$-dimensional submanifold $M \subseteq \mathbb R^d$ can be approximated by rationals also lying on $M$, establishing an upper bound on the "intrinsic Dirichlet exponent" for $M$. We show that relative to this exponent, the set of badly intrinsically approximable points is of full dimension and the set of very well intrinsically approximable points is of zero measure. Our bound on the intrinsic Dirichlet exponent is phrased in terms of an explicit function of $k$ and $d$ which does not seem to have appeared in the literature pr…
In the present paper we establish bounds for the size of the spectral gap for actions of algebraic groups on certain homogeneous spaces. Our approach is based on estimating operator norms of suitable averaging operators, and we develop techniques for establishing both upper and lower bounds for such norms. We shall show that this analytic problem is closely related to the arithmetic problem of establishing bounds on the discrepancy of distribution for rational points on algebraic group varieties. As an application, we show how to establish an effective bound for property $τ$ of congruence subg…
We study a norm sensitive Diophantine approximation problem arising from the work of Davenport and Schmidt on the improvement of Dirichlet's theorem. Its supremum norm case was recently considered by the first-named author and Wadleigh, and here we extend the set-up by replacing the supremum norm with an arbitrary norm. This gives rise to a class of shrinking target problems for one-parameter diagonal flows on the space of lattices, with the targets being neighborhoods of the critical locus of a suitably scaled norm ball. We use methods from geometry of numbers and dynamics to generalize a res…
If $G$ is a compact group acting continuously on a compact metric space $(X, m)$, we prove two results that generalize Dirichlet's classical theorem on Diophantine approximation. If $G$ is a noncommutative compact group of isometries, we obtain a noncommutative form of Dirichlet's theorem. We apply our general result to the special case of the unitary group $U(N)$ acting on the complex unit sphere, and obtain a noncommutative result in this setting.
We give a necessary and sufficient condition for the following property of an integer $d\in\mathbb N$ and a pair $(a,A)\in\mathbb R^2$: There exist $κ> 0$ and $Q_0\in\mathbb N$ such that for all $\mathbf x\in \mathbb R^d$ and $Q\geq Q_0$, there exists $\mathbf p/q\in\mathbb Q^d$ such that $1\leq q\leq Q$ and $\|\mathbf x - \mathbf p/q\| \leq κq^{-a} Q^{-A}$. This generalizes Dirichlet's theorem, which states that this property holds (with $κ= Q_0 = 1$) when $a = 1$ and $A = 1/d$. We also analyze the set of exceptions in those cases where the statement does not hold, showing that they form a co…
We prove $S$-arithmetic inhomogeneous Khintchine type theorems on analytic nondegenerate manifolds. The divergence case, which constitutes the main substance of this paper, is proved in the general context of Hausdorff measures using ubiquitous systems. For $S$ consisting of more than one prime, finite or infinite, the divergence results are new even in the homogeneous setting. We also prove the convergence case of the theorem, including in particular, the $S$-arithmetic inhomogeneous counterpart of the Baker-Sprindžuk conjectures using nondivergence estimates for flows on homogeneous spaces i…
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…
For a decreasing real valued function $ψ$, a pair $(A,\mathbf{b})$ of a real $m\times n$ matrix $A$ and $\mathbf{b}\in\mathbb{R}^m$ is said to be $ψ$-Dirichlet improvable if the system $$\|A\mathbf{q}+\mathbf{b}-\mathbf{p}\|^m < ψ(T)\quad\text{and}\quad\|\mathbf{q}\|^n < T$$ has a solution $\mathbf{p}\in\mathbb{Z}^m$, $\mathbf{q}\in\mathbb{Z}^n$ for all sufficiently large $T$, where $\|\cdot\|$ denotes the supremum norm. Kleinbock and Wadleigh (2019) established an integrability criterion for the Lebesgue measure of the $ψ$-Dirichlet non-improvable set. In this paper, we prove a similar criter…
We extend the Khintchine transference inequalities, as well as a homogeneous-inhomogeneous transference inequality for lattices, due to Bugeaud and Laurent, to a weighted setting. We also provide applications to inhomogeneous Diophantine approximation on manifolds and to weighted badly approximable vectors. Finally, we interpret and prove a conjecture of Beresnevich-Velani (2010) about inhomogeneous intermediate exponents.
