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Dmitry Kleinbock

Publications and source records attributed to Dmitry Kleinbock.

At least 19 recordsLinked to original sources
arXiv2026

Density of the multidimensional Lagrange spectrum

The Lagrange spectrum is a classical object in number theory, defined as the set of values of $\liminf_{q\to\infty} q\, \mathrm{dist}(q\alpha,\mathbb{Z})$ where $\alpha$ runs through irrational numbers. It has a complicated structure, with the discrete part, Hall's ray, and a transitional part in between. One can similarly define Lagrange spectrum in the multidimensional set-up, and until now not much has been understood about it. In this paper we prove that, unlike in the one-dimensional case, the closure of the multidimensional Lagrange spectrum is equal to the interval between $0$ and its supremum. The proof relies on a correspondence between Diophantine approximation and dynamics on the space of unimodular lattices and proceeds by studying a dynamical counterpart of the Lagrange spectrum that we call dynamical Lagrange spectrum. The latter is shown to be equal to the interval between $0$ and its maximum by means of an argument utilizing the higher rank nature of the set-up. A passage from full dynamical spectrum to the density of the Diophantine spectrum is achieved by applying equidistribution of expanding translates of horospheres in the space of lattices.

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arXiv2026

Bounded trajectories of quasi-rays on homogeneous spaces and Diophantine approximation with weight functions

Let $G$ be a connected semisimple real Lie group, $\Gamma$ an irreducible lattice in $G$ and $X = G/\Gamma$. Let $F = \{g_t: t\ge 0\}$ be a non-quasiunipotent one-parameter subsemigroup of $G$. Then it is known that the set of points in $X$ with bounded $F$-trajectories has full Hausdorff dimension. In addition, if $U$ is the expanding horospherical subgroup relative to $g_1$, then for any $x \in X$ the set of points $u \in U$ such that the $F$-trajectory of $ux$ is bounded has full Hausdorff dimension. In this paper we take $U$ to be a horospherical subgroup of $G$ and apply Shi's equidistribution theorem for elements of the expanding cone with respect to $U$ to describe a class of subsets $F$ in $G$, not presupposing the group structure, for which the above full Hausdorff dimension statements also hold. As an application, we prove that the set of badly approximable matrices in the set-up of Diophantine approximations with quasimultiplicative weight functions has full Hausdorff dimension.

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arXiv2026

Uniform Diophantine approximation with restrictions via total density of collections of subspaces

In 1926 Khintchine introduced a topological argument proving the existence of uncountably many nontrivial singular linear forms of $n \geq 2$ variables. Throughout the years, this argument has been extensively modified and generalized. Most recently, Kleinbock et al. (2025) introduced a general framework of Diophantine systems and showed that a certain topological property called total density implies a far-reaching generalization of Khintchine's result. We describe a way to establish total density for a variety of Diophantine systems, and thus prove that the sets of singular objects are uncountable and dense in a wide range of set-ups in Diophantine approximation. As a special case, we establish such a result for inhomogeneous approximation, proving the existence of uncountably many singular systems of affine forms with a fixed translation part. One can also consider approximation with prime denominators, or more generally, approximation under some strong restrictions on numerators and denominators.

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arXiv2025

Constructing bounded orbits of special types on homogeneous spaces

Let $X = G/\Gamma$ be a quotient of a real Lie group by a non-uniform lattice. Consider a one-parameter subgroup $F$ of $G$ that is $\operatorname{Ad}$-diagonalizable over $\mathbb{C}$ and whose action on $(X,m_X)$ is mixing. In this dynamical system we study the set of points $x \in X$ with a precompact orbit, written as $E(F,\infty)$, which is known to be a dense subset of $X$ of full Hausdorff dimension. We prove that $E(F,\infty)$ is indecomposable in the following sense: given any $y \in E(F,\infty)$, the set of $x \in E(F,\infty)$ for which $y \in \overline{F_+x}$, where $F_+$ denotes the positive ray in $F$, is uncountable and dense in $E(F,\infty)$. When the dimension of the neutral subgroup of $G$ with respect to $F$ is $1$ we demonstrate, for any $\varepsilon>0$, the existence of many points $x \in X$ whose orbit closure $\overline{F_+x} \subset X$ is compact and has Hausdorff dimension at least $\dim X - \varepsilon$.

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arXiv2025

Simultaneously bounded and dense orbits for commuting Cartan actions

In this paper we prove that the set of points that have bounded orbits under one regular diagonal flow and dense orbits under the other diagonal flow commuting with the first one has full Hausdorff dimension in $X_3=\mathrm{SL}_3(\mathbb{R})/\mathrm{SL}_3(\mathbb{Z})$. To explain its application towards the Uniform Littlewood's Conjecture proposed in \cite{BFK25}, we introduce the concept of ``fiberwise nondivergence'' for the action of a cone inside the full diagonal subgroup. Then our main result implies that there exists a dense subset of $X_3$ in which each point has a fiberwise non-divergent orbit under a cone inside the full diagonal subgroup and an unbounded orbit under every diagonal flow.

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arXiv2025

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

In this paper, we prove a new ergodic theorem for $\mathbb{R}^d$-actions involving averages over dilated submanifolds, thereby generalizing the theory of spherical averages. Our main result is a quantitative estimate for the error term of such averages valid for smooth functions under some effective mixing assumptions on the action. With the aid of this theorem, we investigate multiplicative-type Dirichlet-improvability for $(m\times n)$-matrices with real coefficients. In particular, we establish that almost all matrices are uniformly approximable by the function $x\mapsto x^{-1}(\log x)^{-1+\varepsilon}$ for any $\varepsilon>0$. Results of this type motivate a question which can be thought as a strengthening of Littlewood's conjecture in multiplicative Diophantine approximation.

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arXiv2024

Singularity, weighted uniform approximation, intersections and rates

A classical argument was introduced by Khintchine in 1926 in order to exhibit the existence of totally irrational singular linear forms in two variables. This argument was subsequently revisited and extended by many authors. For instance, in 1959 Jarnik used it to show that for $n \geq 2$ and for any non-increasing positive $f$ there are totally irrational matrices $A \in M_{m,n}({\mathbb R})$ such that for all large enough $t$ there are $\mathbf{p} \in {\mathbb Z}^m, \mathbf{q} \in {\mathbb Z}^n \smallsetminus \{0\}$ with $$\|\mathbf{q}\| \leq t \ \text{ and } \ \|A \mathbf{q} - \mathbf{p}\| \leq f(t).$$ We denote the collection of such matrices by $\mathrm{UA}^*_{m,n}(f)$. We adapt Khintchine's argument to show that the sets $\mathrm{UA}^*_{m,n}(f)$, and their weighted analogues $\mathrm{UA}^*_{m,n}(f, \mathbf{w})$, intersect many manifolds and fractals, and have strong intersection properties. For example, we show that: When $n \geq 2$, the set $\bigcap_{\mathbf{w}} \mathrm{UA}^*(f, \mathbf{w}) $, where the intersection is over all weights $\mathbf{w}$, is nonempty, and moreover intesects many manifolds and fractals; For $n \geq 2$, there are vectors in ${\mathbb R}^n$ which are simultaneously $k$-singular for every $k$, in the sense of Yu; when $n \geq 3$, $\mathrm{UA}^*_{1,n}(f) + \mathrm{UA}^*_{1,n}(f) = {\mathbb R}^n$. We also obtain new bounds on the rate of singularity which can be attained by column vectors in analytic submanifolds of dimension at least 2 in ${\mathbb R}^n$.

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arXiv2023

Dimension bounds for escape on average in homogeneous spaces

Let $X = G/\Gamma$, where $G$ is a Lie group and $\Gamma$ is a uniform lattice in $G$, and let $O$ be an open subset of $X$. We give an upper estimate for the Hausdorff dimension of the set of points whose trajectories escape $O$ on average with frequency $\delta$, where $0 < \delta \le 1$.

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arXiv2022

On Multiplicatively Badly Approximable Vectors

Let $\langle x\rangle$ denote the distance from $x\in\mathbb{R}$ to the set of integers $\mathbb{Z}$. The Littlewood Conjecture states that for all pairs $(\alpha,\beta)\in\mathbb{R}^{2}$ the product $q\langle q\alpha\rangle\langle q\beta\rangle$ attains values arbitrarily close to $0$ as $q\in\mathbb{N}$ tends to infinity. Badziahin showed that if a factor $\log q\cdot \log\log q$ is added to the product, the same statement becomes false. In this paper, we generalise Badziahin's result to vectors $\boldsymbol{\alpha}\in\mathbb{R}^{d}$, replacing the function $\log q\cdot \log\log q$ by $(\log q)^{d-1}\cdot\log\log q$ for any $d\geq 2$, and thereby obtaining a new proof in the case $d=2$. Our approach is based on a new version of the well-known Dani Correspondence between Diophantine approximation and dynamics on the space of lattices, especially adapted to the study of products of rational approximations. We believe that this correspondence is of independent interest.

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arXiv2022

A dichotomy phenomenon for Bad minus normed Dirichlet

Given a norm $\nu$ on $\mathbb{R}^2$, the set of $\nu$-Dirichlet improvable numbers $\mathbf{DI}_\nu$ was defined and studied in the papers of Andersen-Duke (Acta Arith. 2021) and Kleinbock-Rao (Internat. Math. Res. Notices 2022). When $\nu$ is the supremum norm, $\mathbf{DI}_\nu = \mathbf{BA}\cup \mathbb{Q}$, where $\mathbf{BA}$ is the set of badly approximable numbers. Each of the sets $\mathbf{DI}_\nu$, like $\mathbf{BA}$, is of measure zero and satisfies the winning property of Schmidt. Hence for every norm $\nu$, $\mathbf{BA} \cap \mathbf{DI}_\nu$ is winning and thus has full Hausdorff dimension. In the present article we prove the following dichotomy phenomenon: either $\mathbf{BA} \subset \mathbf{DI}_\nu$ or else $\mathbf{BA} \smallsetminus \mathbf{DI}_\nu$ has full Hausdorff dimension. We give several examples for each of the two cases. The dichotomy is based on whether the critical locus of $\nu$ intersects a precompact $g_t$-orbit, where $\{g_t\}$ is the one-parameter diagonal subgroup of $\operatorname{SL}_2(\mathbb{R})$ acting on the space $X$ of unimodular lattices in $\mathbb{R}^2$. Thus the aforementioned dichotomy follows from the following dynamical statement: for a lattice $\Lambda\in X$, either $g_\mathbb{R} \Lambda$ is unbounded (and then any precompact $g_{\mathbb{R}_{>0}}$-orbit must eventually avoid a neighborhood of $\Lambda$), or not, in which case the set of lattices in $X$ whose $g_{\mathbb{R}_{>0}}$-trajectories are precompact and contain $\Lambda$ in their closure has full Hausdorff dimension.

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arXiv2022

Dimension drop for diagonalizable flows on homogeneous spaces

Let $X = G/\Gamma$, where $G$ is a Lie group and $\Gamma$ is a lattice in $G$, let $O$ be an open subset of $X$, and let $F = \{g_t: t\ge 0\}$ be a one-parameter subsemigroup of $G$. Consider the set of points in $X$ whose $F$-orbit misses $O$; it has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of $X$. This conjecture is proved when $X$ is compact or when $G$ is a simple Lie group of real rank $1$, or, most recently, for certain special flows on the space of lattices. In this paper we prove this conjecture for arbitrary $\operatorname{Ad}$-diagonalizable flows on irreducible quotients of semisimple Lie groups. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on $G/\Gamma$. We also derive an application to jointly Dirichlet-Improvable systems of linear forms.

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arXiv2022

Dynamical Borel-Cantelli Lemma for Recurrence under Lipschitz Twists

In the study of some dynamical systems the limsup set of a sequence of measurable sets is often of interest. The shrinking targets and recurrence are two of the most commonly studied problems that concern limsup sets. However, the zero-one laws for the shrinking targets and recurrence are usually treated separately and proved differently. In this paper, we introduce a generalized definition that can specialize into the shrinking targets and recurrence; our approach gives a unified proof of the zero-one laws for the two problems.

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arXiv2022

Inhomogeneous Diophantine approximation for generic homogeneous functions

The present paper is a sequel to [Monatsh.~Math.\ {\bf 194} (2021), 523--554] in which results of that paper are generalized so that they hold in the setting of inhomogeneous Diophantine approximation. Given any integers $n \geq 2$ and $\ell \geq 1$, any ${\pmb \xi} = \left(\xi_1, \dots , \xi_\ell \right) \in \mathbb{R}^\ell$, and any homogeneous function \linebreak $f = \left(f_1, \dots , f_\ell \right): \mathbb{R}^n \to \mathbb{R}^\ell$ that satisfies a certain nonsingularity assumption, we obtain a biconditional criterion on the approximating function $\psi = \left(\psi_1, \dots , \psi_\ell \right): \mathbb{R}_{\geq 0} \to \left(\mathbb{R}_{>0}\right)^\ell$ for a generic element $f \circ g$ in the $\operatorname{SL}_n(\mathbb{R})$-orbit of $f$ to be (respectively, not to be) $\psi$-approximable at ${\pmb \xi} = (\xi_1,\dots,\xi_n)$: that is, for there to exist infinitely many (respectively, only finitely many) $\mathbf{v} \in \mathbb{Z}^n$ such that $\left|\xi_j - \left( f_j \circ g\right)(\mathbf{v})\right| \leq \psi_j(\|\mathbf{v}\|)$ for each $j \in \left\lbrace 1, \dots, \ell \right\rbrace$. In this setting, we also obtain a sufficient condition for uniform approximation. We also consider some examples of $f$ that do not satisfy our nonsingularity assumptions and prove similar results for these examples. Moreover, one can replace $\operatorname{SL}_n(\mathbb{R})$ above by any closed subgroup of $\operatorname{ASL}_n(\mathbb{R})$ that satisfies certain integrability axioms (being of Siegel and Rogers type) introduced by the authors in the aforementioned previous paper.

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arXiv2022

Metrical properties for the weighted products of multiple partial quotients in continued fractions

The classical Khintchine and Jarn\'ik theorems, generalizations of a consequence of Dirichlet's theorem, are fundamental results in the theory of Diophantine approximation. These theorems are concerned with the size of the set of real numbers for which the partial quotients in their continued fraction expansions grows with a certain rate. Recently it was observed that the growth of product of pairs of consecutive partial quotients in the continued fraction expansion of a real number is associated with improvements to Dirichlet's theorem. In this paper we consider the products of several consecutive partial quotients raised to different powers. Namely, we find the Lebesgue measure and the Hausdorff dimension of the following set: $$ {\D_{\mathbf t}}(\psi):=\left\{x\in[0, 1): \prod\limits_{i=0}^{m-1}{a^{t_i}_{n+i}(x)} \ge \Psi(n)\ {\text{for infinitely many}} \ n\in \N \right\}, $$ where $t_i\in\mathbb R_+$ for all ${0\leq i\leq m-1}$, and $\Psi:\N\to\R_{\ge 1}$ is a positive function.

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arXiv2021

Weighted uniform Diophantine approximation of systems of linear forms

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 the corresponding norm. The trajectory is formed by the action of a one-parameter diagonal subgroup corresponding to the weights. We first state a very precise form of Dirichlet's theorem and prove it for some norms. Secondly we show, for these same norms, that the set of Dirichlet-improvable matrices has full Hausdorff dimension. Though the techniques used vary greatly depending on the chosen norm, we expect these results to hold in general.

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arXiv2021

A measure estimate in geometry of numbers and improvements to Dirichlet's theorem

Let $\psi$ be a continuous decreasing function defined on all large positive real numbers. We say that a real $m\times n$ matrix $A$ is $\psi$-Dirichlet if for every sufficiently large real number $t$ one can find $\boldsymbol{p} \in \mathbb{Z}^m$, $\boldsymbol{q} \in \mathbb{Z}^n\smallsetminus\{\boldsymbol{0}\}$ satisfying $\|A\boldsymbol{q}-\boldsymbol{p}\|^m< \psi({t})$ and $\|\boldsymbol{q}\|^n<{t}$. This property was introduced by Kleinbock and Wadleigh in 2018, generalizing the property of $A$ being Dirichlet improvable which dates back to Davenport and Schmidt (1969). In the present paper, we give sufficient conditions on $\psi$ to ensure that the set of $\psi$-Dirichlet matrices has zero or full Lebesgue measure. Our proof is dynamical and relies on the effective equidistribution and doubly mixing of certain expanding horospheres in the space of lattices. Another main ingredient of our proof is an asymptotic measure estimate for certain compact neighborhoods of the critical locus (with respect to the supremum norm) in the space of lattices. Our method also works for the analogous weighted problem where the relevant supremum norms are replaced by certain weighted quasi-norms.

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arXiv2021

Abundance of Dirichlet-improvable pairs with respect to arbitrary norms

In a recent paper of Akhunzhanov and Shatskov the two-dimensional Dirichlet spectrum with respect to Euclidean norm was defined. We consider an analogous definition for arbitrary norms on $\mathbb{R}^2$ and prove that, for each such norm, the set of Dirichlet improvable pairs contains the set of badly approximable pairs, hence is hyperplane absolute winning. To prove this we make a careful study of some classical results in the geometry of numbers due to Chalk--Rogers and Mahler to establish a Haj\'{o}s--Minkowski type result for the critical locus of a cylinder. As a corollary, using a recent result of the first named author with Mirzadeh, we conclude that for any norm on $\mathbb{R}^2$ the top of the Dirichlet spectrum is not an isolated point.

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