arXiv subjects
Publications and source records attributed to Sanju Velani.
In this paper we discuss a general problem on metrical Diophantine approximation associated with a system of linear forms. The main result is a zero-one law that extends one-dimensional results of Cassels and Gallagher. The paper contains a discussion on possible generalisations including a selection of various open problems.
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 inhomogeneous theory contains the homogeneous theory and so is more general. As a consequence, we establish the inhomogeneous analogue of the Baker-Sprindzuk conjecture. Furthermore, we prove a complete inhomogeneous version of the profound theorem of Kleinbock, Lindenstrauss & Weiss on the extremality of friendly measures. The results obtained in this paper constitute the first step towards developing a coherent inhomogeneous theory for manifolds in line with the homogeneous theory.
In 1998, Kleinbock & Margulis established a conjecture of V.G. Sprindzuk in metrical Diophantine approximation (and indeed the stronger Baker-Sprindzuk conjecture). In essence the conjecture stated that the simultaneous homogeneous Diophantine exponent $w_{0}(\vv x) = 1/n$ for almost every point $\vv x$ on a non-degenerate submanifold $\cM$ of $\R^n$. In this paper the simultaneous inhomogeneous analogue of Sprindzuk's conjecture is established. More precisely, for any `inhomogeneous' vector $\bmθ\in\R^n$ we prove that the simultaneous inhomogeneous Diophantine exponent $w_{0}(\vv x, \bmθ)= 1/n$ for almost every point $\vv x$ on $M$. The key result is an inhomogeneous transference principle which enables us to deduce that the homogeneous exponent $w_0(\vv x)=1/n$ for almost all $\vv x\in \cM$ if and only if for any $\bmθ\in\R^n$ the inhomogeneous exponent $w_0(\vv x,\bmθ)=1/n$ for almost all $\vv x\in \cM$. The inhomogeneous transference principle introduced in this paper is an extremely simplified version of that recently discovered in \cite{Beresnevich-Velani-new-inhom}. Nevertheless, it should be emphasised that the simplified version has the great advantage of bringing to the forefront the main ideas of \cite{Beresnevich-Velani-new-inhom} while omitting the abstract and technical notions that come with describing the inhomogeneous transference principle in all its glory.
There are two fundamental results in the classical theory of metric Diophantine approximation: Khintchine's theorem and Jarnik's theorem. The former relates the size of the set of well approximable numbers, expressed in terms of Lebesgue measure, to the behavior of a certain volume sum. The latter is a Hausdorff measure version of the former. We start by discussing these theorems and show that they are both in fact a simple consequence of the notion of `local ubiquity'. The local ubiquity framework introduced here is a much simplified and more transparent version of that in \cite{memoirs}. Furthermore, it leads to a single local ubiquity theorem that unifies the Lebesgue and Hausdorff theories. As an application of our framework we consider the theory of metric Diophantine approximation on limit sets of Kleinian groups. In particular, we obtain a general Hausdorff measure version of Sullivan's logarithm law for geodesics -- an aspect overlooked in \cite{memoirs}.
A Hausdorff measure version of the Duffin-Schaeffer conjecture in metric number theory is introduced and discussed. The general conjecture is established modulo the original conjecture. The key result is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for lim sup subsets of R^k to Hausdorff measure theoretic statements. In view of this, the Lebesgue theory of lim sup sets is shown to underpin the general Hausdorff theory. This is rather surprising since the latter theory is viewed to be a subtle refinement of the former.
Let $\cal C$ be a non--degenerate planar curve and for a real, positive decreasing function $ψ$ let $\cal C(ψ)$ denote the set of simultaneously $ψ$--approximable points lying on $\cal C$. We show that $\cal C$ is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on $\cal C$ of $\cal C(ψ)$ is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. In the case that $\cal C$ is a rational quadric the convergence counterparts of the divergent results are also obtained. Furthermore, for functions $ψ$ with lower order in a critical range we determine a general, exact formula for the Hausdorff dimension of $\cal C(ψ)$. These results constitute the first precise and general results in the theory of simultaneous Diophantine approximation on manifolds.
A Hausdorff measure version of W.M. Schmidt's inhomogeneous, linear forms theorem in metric number theory is established. The key ingredient is a `slicing' technique motivated by a standard result in geometric measure theory. In short, `slicing' together with the Mass Transference Principle [3] allows us to transfer Lebesgue measure theoretic statements for limsup sets associated with linear forms to Hausdorff measure theoretic statements. This extends the approach developed in [3] for simultaneous approximation. Furthermore, we establish a new Mass Transference Principle which incorporates both forms of approximation. As an application we obtain a complete metric theory for a `fully' non-linear Diophantine problem within the linear forms setup. [3] V. Beresnevich and S. Velani : A Mass Transference Principle and the Duffin--Schaeffer conjecture for Hausdorff measures, Pre-print (22pp): arkiv:math.NT/0401118. To appear: Annals of Math.
Let $K$ denote the middle third Cantor set and ${\cal A}:= \{3^n : n = 0,1,2, >... \} $. Given a real, positive function $ψ$ let $ W_{\cal A}(ψ)$ denote the set of real numbers $x$ in the unit interval for which there exist infinitely many $(p,q) \in \Z \times {\cal A} $ such that $ |x - p/q| < ψ(q) $. The analogue of the Hausdorff measure version of the Duffin-Schaeffer conjecture is established for $ W_{\cal A}(ψ) \cap K $. One of the consequences of this is that there exist very well approximable numbers, other than Liouville numbers, in $K$ -- an assertion attributed to K. Mahler.
Let (X,d) be a metric space and (Ω, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of Ω. Loosely speaking, these consist of points in Ωwhich `stay clear' of some given set of points in X. The classical set \Bad of `badly approximable' numbers in the theory of Diophantine approximation falls within our framework as do the sets \Bad(i,j) of simultaneously badly approximable numbers. Under various natural conditions we prove that the badly approximable subsets of Ωhave full Hausdorff dimension. Applications of our general framework include those from number theory (classical, complex, p-adic and formal power series) and dynamical systems (iterated function schemes, rational maps and Kleinian groups).
Let $\cS_n(ψ_1,...,ψ_n)$ denote the set of simultaneously $(ψ_1,...,ψ_n)$--approximable points in $\R^n$ and $\cSM_n(ψ)$ denote the set of multiplicatively $ψ$--approximable points in $\R^n$. Let $\cM$ be a manifold in $\R^n$. The aim is to develop a metric theory for the sets $ \cM \cap \cS_n(ψ_1,...,ψ_n) $ and $ \cM \cap \cSM_n(ψ) $ analogous to the classical theory in which $\cM$ is simply $\R^n$. In this note, we mainly restrict our attention to the case that $\cM$ is a planar curve $\cC$. A complete Hausdorff dimension theory is established for the sets $ \cC \cap \cS_2(ψ_1,ψ_2) $ and $ \cC \cap \cSM_2(ψ) $. A divergent Khintchine type result is obtained for $\cC \cap \cS_2(ψ_1,ψ_2) $; i.e. if a certain sum diverges then the one--dimensional Lebesgue measure on $\cC$ of $\cC \cap \cS_2(ψ_1,ψ_2) $ is full. Furthermore, in the case that $\cC$ is a rational quadric the convergent Khintchine type result is obtained for both types of approximation. Our results for $\cC \cap \cS_2(ψ_1,ψ_2) $ naturally generalize the dimension and Lebesgue measure statements of \cite{BDV03}. Within the multiplicative framework, our results for $ \cC \cap \cSM_2(ψ)$ constitute the first of their type.
Let $W(\p)$ denote the set of $\p$-well approximable points in $\R^d$ and let $K$ be a compact subset of $\R^d$ which supports a measure $μ$. In this short note, we show that if $μ$ is an `absolutely friendly' measure and a certain $μ$--volume sum converges then $μ(W(\p) \cap K) = 0$. The result obtained is in some sense analogous to the convergence part of Khintchines classical theorem in the theory of metric Diophantine approximation. The class of absolutely friendly measures is a subclass of the friendly measures introduced by D. Kleinbock, E. Lindenstrauss and B. Weiss (On fractal measures and Diophantine approximation) and includes measures supported on self similar sets satisfying the open set condition. We also obtain an upper bound result for the Hausdorff dimension of $W(\p) \cap K $.
Given a compact metric space (X,d) equipped with a non-atomic, probability measure m and a real, positive decreasing function p we consider a `natural' class of limsup subsets La(p) of X. The classical limsup sets of `well approximable' numbers in the theory of metric Diophantine approximation fall within this class. We show that m(La(p))>0 under a `global ubiquity' hypothesis and the divergence of a certain m--volume sum. In fact, under a `local ubiquity' hypothesis we show that La(p) has full measure; i.e. m(La(p)) =1 . This is the analogue of the divergent part of the classical Khintchine-Groshev theorem in number theory. Moreover, if the 'local ubiquity' hypothesis is satisfied and a certain f-volume sum diverges then we are able to show that the Hausdorff f--measure of La(p) is infinite. A simple consequence of this is a lower bound for the Hausdorff dimension of La(p) and various results concerning the dimension and measure of related `exact order' sets. Essentially, the notion of `local ubiquity' unexpectedly unifies `divergent' type results for La(p) with respect to the natural measure m and general Hausdorff measures. Applications of the general framework include those from number theory, Kleinian groups and rational maps. Even for the classical limsup sets of `well approximable' numbers, the framework strengthens the classical Hausdorff measure result of Jarnik and opens up the Duffin-Schaeffer conjecture for Hausdorff measures.