V. I. Bernik

Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions

Notation. The main objects of this paper are n-tuples y = (y1, . . . , yn) of real numbers viewed as linear forms, i.e. as row vectors. In what follows, y will always mean a row vector, and we will be interested in values of a linear form given by y at integer points q = (q1, . . . , qn)T , the latter being a column vector. Thus yq will stand for y1q1 + · · ·+ ynqn. Hopefully it will cause no confusion. We will study differentiable maps f = (f1, . . . , fn) from open subsets U of R to R; again, f will be interpreted as a row vector, so that f(x)q stands for q1f1(x) + · · · + qnfn(x). In contrast, the elements of the “parameter set” U will be denoted by x = (x1, . . . , xd) without boldfacing, since the linear structure of the parameter space is not significant. For f as above we will denote by ∂if : U 7→ R, i = 1, . . . , d, its partial derivative (also a row vector) with respect to xi. If F is a scalar function on U , we will denote by∇F the column vector consisting of partial derivatives of F . With some abuse of notation, the same way we will treat vector functions f : namely, ∇f will stand for the matrix function U 7→ Md×n(R) with rows given by partial derivatives ∂if . We will also need higher order differentiation: for a multiindex β = (i1, . . . , id), ij ∈ Z+, we let |β| = ∑d j=1 ij and ∂β = ∂ i1 1 ◦ · · · ◦ ∂ id d . Unless otherwise indicated, the norm ‖x‖ of a vector x ∈ R (either row or column vector) will stand for ‖x‖ = max1≤i≤k |xi|. In some cases however we will work with the Euclidean norm ‖x‖ = ‖x‖e = √∑k i=1 x 2 i , keeping the same notation. This distinction will be clearly emphasized to avoid confusion. We will denote by R1 the set of unit vectors in R (with respect to the Euclidean norm). We will use the notation |〈x〉| for the distance between x ∈ R and the closest integer, |〈x〉| def = mink∈Z |x− k|. (It is quite customary to use ‖x‖ instead, but we are not going to do this in order to save the latter notation for norms in vector spaces.) If B ⊂ R, we let |B| stand for the Lebesgue measure of B.

The distribution of close conjugate algebraic numbers

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.

Simultaneous Diophantine approximation in the real, complex and p –adic fields.

In this paper it is shown that if the volume sum ?r = 18 ?(r) converges for a monotonic function ? then the set of points (x, z, w) ? R × C × Qp which simultaneously satisfy the inequalities |P(x)| = H-v1 ??1(H), |P(z)| = H-v2 ??2(H) and |P(w)|p = H-v3 ??3(H) with v1 + 2v2 + v3 = n - 3 and ?1 + 2?2 + ?3 = 1 for infinitely many integer polynomials P has measure zero.

The Khintchine–Groshev theorem for planar curves

The analogue of the classical Khintchine–Groshev theorem, a fundamental result in metric Diophantine approximation, is established for smooth planar curves with non–vanishing curvature almost everywhere.

Distribution of Algebraic Numbers and Metric Theory of Diophantine Approximation

In this paper we give an overview of recent results regarding close conjugate algebraic numbers, the number of integral polynomials with small discriminant and pairs of polynomials with small resultants.

A multipoint problem with multiple nodes for linear hyperbolic equations

We establish conditions for the unique solvability of a multipoint (with respect to the time coordinate) problem with multiple nodes for linear hyperbolic equations with constant coefficients in the class of functions periodic in the space variable. We prove metric statements concerning lower bounds of small denominators that appear in the course of construction of a solution of the problem.

On the Distribution of Points with Algebraically Conjugate Coordinates in a Neighborhood of Smooth Curves

Let φ : ℝ → ℝ be a continuously differentiable function on a finite interval J ⊂ ℝ, and let α = (α1, α2) be a point with algebraically conjugate coordinates such that the minimal polynomial P of α1, α2 is of degree ≤ n and height ≤ Q. Denote by MφnQγJ

Metric diophantine approximation on manifolds

{M}_{\varphi}^n\left(Q,\gamma, J\right)