arXiv subjects
Publications and source records attributed to Anish Ghosh.
We resolve problems posed by Kesten and Erdös-Szüsz-Turán on probabilistic Diophantine approximation via methods of homogeneous dynamics. Our methods allows us to generalize the problem to the setting of general measure-valued processes in $\mathbb{R}^n$, and obtain applications to the distribution of point sets which occur in higher dimensional Diophantine approximation and the geometry of translation surfaces.
This is a survey article describing some recent results at the interface of homogeneous dynamics and Diophantine approximation.
We study some problems in metric Diophantine approximation over local fields of positive characteristic.
The overall aim of this note is to initiate a "manifold" theory for metric Diophantine approximation on the limit sets of Kleinian groups. We investigate the notions of singular and extremal limit points within the geometrically finite Kleinian group framework. Also, we consider the natural analogue of Davenport's problem regarding badly approximable limit points in a given subset of the limit set. Beyond extremality, we discuss potential Khintchine-type statements for subsets of the limit set. These can be interpreted as the conjectural "manifold" strengthening of Sullivan's logarithmic law for geodesics.
We establish that the optimal bound for the size of the smallest integral solution of the Oppenheim Diophantine approximation problem $\abs{Q(x)-ξ}< ε$ for a generic ternary form $Q$ is $\abs{x}\ll ε^{-1}$. We also establish an optimal rate of density for the values of polynomials maps in a number of other natural problems, including the values of linear forms restricted to suitable quadratic surfaces, and the values of the polynomial map defined by the generators of the ring of conjugation-invariant polynomials on $M_3(\C)$. These results are instances of a general approach that we develop, which considers a rational affine algebraic subvariety of Euclidean space, invariant and homogeneous under an action of a semisimple Lie group $G$. Given a polynomial map $F$ defined on the Euclidean space which is invariant under a semisimple subgroup $H$ of the acting group $G$, consider the family of its translates $F\circ g$ by elements of the group. We study the restriction of these polynomial functions to the integer points on the variety confined to a large Euclidean ball. Our main results establish an explicit rate of density for their values, for generic polynomials in the family. This problem has been extensively studied before when the polynomials in question are linear, in the context of classical Diophantine approximation, but very little was known about it for polynomial of higher degree. We formulate a heuristic pigeonhole lower bound for the density and an explicit upper bound for it, formulate a sufficient condition for the coincidence of the lower and upper bounds, and in a number of natural examples establish that they indeed match. Finally, we also establish a rate of density for values of homogeneous polynomials on homogeneous projective varieties.
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.
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 provide an extension of the transference results of Beresnevich and Velani connecting homogeneous and inhomogeneous Diophantine approximation on manifolds and provide bounds for inhomogeneous Diophantine exponents of affine subspaces and their nondegenerate submanifolds.
We prove a quantitative version of Oppenheim's conjecture for generic ternary indefinite quadratic forms. Our results are inspired by and analogous to recent results for diagonal quadratic forms due to Bourgain.
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 study the problem of Diophantine approximation on lines in $\mathbb{R}^d$ under certain primality restrictions.
We study the shrinking target problem for actions of semisimple groups on homogeneous spaces, with applications to logarithm laws and Diophantine approximation.
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.
We investigate the geometry of approximates in multiplicative Diophantine approximation. Our main tool is a new multiparameter averaging result for Siegel transforms on the space of unimodular lattices in ${\mathbb R}^n$ which is of independent interest.
We consider the question of how approximations satisfying Dirichlet's theorem spiral around vectors in $\mathbb{R}^d$. We give pointwise almost everywhere results (using only the Birkhoff ergodic theorem on the space of lattices). In addition, we show that for $\textit{every}$ unimodular lattice, on average, the directions of approximates spiral in a uniformly distributed fashion on the $d-1$ dimensional unit sphere. For this second result, we adapt a very recent proof of Marklof and Strömbergsson \cite{MS3} to show a spherical average result for Siegel transforms on $\operatorname{SL}_{d+1}(\mathbb{R})/\operatorname{SL}_{d+1}(\mathbb{Z})$. Our techniques are elementary. Results like this date back to the work of Eskin-Margulis-Mozes \cite{EMM} and Kleinbock-Margulis \cite{KM} and have wide-ranging applications. We also explicitly construct examples in which the directions are not uniformly distributed.
The present paper establishes upper and lower bounds on the speed of approximation in a wide range of natural Diophantine approximation problems. The upper and lower bounds coincide in many cases, giving rise to optimal results in Diophantine approximation which were inaccessible previously. Our approach proceeds by establishing, more generally, upper and lower bounds for the rate of distribution of dense orbits of a lattice subgroup $Γ$ in a connected Lie (or algebraic) group $G$, acting on suitable homogeneous spaces $G/H$. The upper bound is derived using a quantitative duality principle for homogeneous spaces, reducing it to a rate of convergence in the mean ergodic theorem for a family of averaging operators supported on $H$ and acting on $G/Γ$. In particular, the quality of the upper bound on the rate of distribution we obtain is determined explicitly by the spectrum of $H$ in the automorphic representation on $L^2(Γ\setminus G)$. We show that the rate is best possible when the representation in question is tempered, and show that the latter condition holds in a wide range of examples.
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 specifically on the problem of estimating exponents of Diophantine approximation by arithmetic lattices acting on algebraic varieties. In the third part of the paper, we exemplify our results by establishing explicit bounds for the Diophantine exponent of dense lattice orbits in a number of basic cases. These include the linear and affine actions on affine spaces, and the action on the variety of matrices of fixed determinant. In some cases, these exponents are shown to be best possible.
We show that points on $C^{1}$ curves which are badly approximable by rationals in a number field form a winning set in the sense of W. M. Schmidt. As a consequence, we obtain a number field version of Schmidt's conjecture.