Victor Beresnevich

A GROSHEV TYPE THEOREM FOR CONVERGENCE ON MANIFOLDS

We deal with Diophantine approximation on the so-called non-degenerate manifolds and prove an analogue of the Khintchine–Groshev theorem. The problem we consider was first posed by A. Baker [1] for the rational normal curve. The non-degenerate manifolds form a large class including any connected analytic manifold which is not contained in a hyperplane. We also present a new approach which develops the ideas of Sprindzuks classical method of essential and inessential domains first used by him to solve Mahlers problem [28].

Badly approximable points on manifolds

This paper is motivated by two problems in the theory of Diophantine approximation, namely, Davenport’s problem regarding badly approximable points on submanifolds of a Euclidean space and Schmidt’s problem regarding the intersections of the sets of weighted badly approximable points. The problems have been recently settled in dimension two but remain open in higher dimensions. In this paper we develop new techniques that allow us to tackle them in full generality. The techniques rest on lattice points counting and a powerful quantitative result of Bernik, Kleinbock and Margulis. The main theorem of this paper implies that any finite intersection of the sets of weighted badly approximable points on any analytic nondegenerate submanifold of

The distribution of close conjugate algebraic numbers

\mathbb {R}^n

On Diophantine Approximations of Dependent Quantities in the p-adic Case

Rn has full dimension. One of the consequences of this result is the existence of transcendental real numbers badly approximable by algebraic numbers of any bounded degree.

The Khintchine–Groshev theorem for planar curves

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.

Distribution of Algebraic Numbers and Metric Theory of Diophantine Approximation

A vehicle for feeding and screening material including a wheel-supported vehicle frame and a material-transport conveyor extending from one end of the vehicle frame to a raised end located between the ends of the vehicle frame. A multiple deck screen assembly forms an extension of the material-transport conveyor, and inclines downwardly toward a discharge end located at the opposite end of the vehicle frame. Off-bearing conveyors are adjustable to transport screened material laterally of the vehicle frame.

A multipoint problem with multiple nodes for linear hyperbolic equations

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.

DIOPHANTINE APPROXIMATION ON MANIFOLDS AND LOWER BOUNDS FOR HAUSDORFF DIMENSION

AbstractIn the present paper, we prove an analog of Khinchins metric theorem in the case of linear Diophantine approximations of plane curves defined over the ring of