arXiv subjects
Publications and source records attributed to Victor Beresnevich.
This paper takes a new step in the direction of proving the Duffin-Schaeffer Conjecture for measures arbitrarily close to Lebesgue. The main result is that under a mild `extra divergence' hypothesis, the conjecture is true.
The inhomogeneous Groshev type theory for dual Diophantine approximation on manifolds is developed. In particular, the notion of nice manifolds is introduced and the divergence part of the theory is established for all such manifolds. Our results naturally incorporate and generalize the homogeneous measure and dimension theorems for non-degenerate manifolds established to date. The generality of the inhomogeneous aspect considered within enables us to make a new contribution even to the classical theory in R^n. Furthermore, the multivariable aspect considered within has natural applications beyond the standard inhomogeneous theory such as to Diophantine problems related to approximation by algebraic integers.
We investigate the distribution of real algebraic numbers of a fixed degree having a close conjugate number, the distance between the conjugate numbers being given as a function of their height. The main result establishes the ubiquity of such algebraic numbers in the real line and implies a sharp quantitative bound on their number. Although the main result is rather general it implies new estimates on the least possible distance between conjugate algebraic numbers, which improve recent bounds of Bugeaud and Mignotte. So far the results a la Bugeaud and Mignotte relied on finding explicit families of polynomials with clusters of roots. Here we suggest a different approach in which irreducible polynomials are implicitly tailored so that their derivatives assume certain values. The applications of our main theorem considered in this paper include generalisations of a theorem of Baker and Schmidt and a theorem of Bernik, Kleinbock and Margulis in the metric theory of Diophantine approximation.
The primary goal of this paper is to complete the theory of metric Diophantine approximation initially developed in [Ann. of Math.(2) 166 (2007), p.367-426] for $C^3$ non-degenerate planar curves. With this goal in mind, here for the first time we obtain fully explicit bounds for the number of rational points near planar curves. Further, introducing a perturbational approach we bring the smoothness condition imposed on the curves down to $C^1$ (lowest possible). This way we broaden the notion of non-degeneracy in a natural direction and introduce a new topologically complete class of planar curves to the theory of Diophantine approximation. In summary, our findings improve and complete the main theorems of [Ann. of Math.(2) 166 (2007), p.367-426] and extend the celebrated theorem of Kleinbock and Margulis appeared in [Ann. of Math.(2), 148 (1998), p.339-360] in dimension 2 beyond the notion of non-degeneracy.
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 unnecessary conditions from the classical Khintchine-Groshev theorem. This settles a multi-dimensional analogue of Catlin's Conjecture.
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 of the distribution of rational points near manifolds -- a very attractive topic in its own right. Here, for the first time, we obtain sharp lower bounds for the number of rational points near non-degenerate manifolds in dimensions n>2 and show that they are ubiquitous (that is uniformly distributed).
The inhomogeneous metric theory for the set of simultaneously $ψ$-approximable points lying on a planar curve is developed. Our results naturally incorporate the homogeneous Khintchine-Jarnik type theorems recently established in [Ann. of Math. (2), 166 (2007), pp. 367-426] and [Invent. Math., 166 (2006), pp. 103-124]. The key lies in obtaining essentially the best possible results regarding the distribution of `shifted' rational points near planar curves.
The idea of using measure theoretic concepts to investigate the size of number theoretic sets, originating with E. Borel, has been used for nearly a century. It has led to the development of the theory of metrical Diophantine approximation, a branch of Number Theory which draws on a rich and broad variety of mathematics. We discuss some recent progress and open problems concerning this classical theory. In particular, generalisations of the Duffin-Schaeffer and Catlin conjectures are formulated and explored.
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.
This paper goes back to a famous problem of Mahler in metrical Diophantine approximation. The problem has been settled by Sprindzuk and subsequently improved by Alan Baker and Vasili Bernik. In particular, Bernik's result establishes a convergence Khintchine type theorem for Diophantine approximation by polynomials, that is it allows arbitrary monotonic error of approximation. In the present paper the monotonicity assumption is completely removed.
A complete p-adic Khintchine type theorem for approximation by p-adic algebraic numbers is established.
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 $\cS_n(ψ_1,...,ψ_n)$ 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.