Related papers: Multiplicative Diophantine approximation and bound…

Lattice sums of cuboidal lattices, which connect the face-centered with the mean-centered and the body-centered cubic lattices through parameter dependent lattice vectors, are evaluated by decomposing them into two separate lattice sums…

Because of its ineffectiveness, the usual arithmetic Hilbert-Samuel formula is not applicable in the context of Diophantine Approximation. In order to overcome this difficulty, the present paper presents explicit estimates for arithmetic…

First, we calculate the Ehrhart polynomial associated to an arbitrary cube with integer coordinates for its vertices. Then, we use this result to derive relationships between the Ehrhart polynomials for regular lattice tetrahedrons and…

We generalise M. M. Skriganov's notion of weak admissibility for lattices to include standard lattices occurring in Diophantine approximation and algebraic number theory, and we prove estimates for the number of lattice points in sets such…

We consider polyhedral approximations of strictly convex compacta in finite dimensional Euclidean spaces (such compacta are also uniformly convex). We obtain the best possible estimates for errors of considered approximations in the…

Considering simultaneous approximation to three numbers, we study the geometry of the sequence of best approximations. We provide a sharper lower bound for the ratio between ordinary and uniform exponent of Diophantine approximation,…

We draw connections between the various conjectures which are included in G. R\'emond's generalized Lehmer problems. Specifically, we show that the degree one form of his conjecture for the multiplicative group is, in a sense, almost as…

These notes represent an extended version of a talk I gave for the participants of the IMO 2009 and other interested people. We introduce diophantine equations and show evidence that it can be hard to solve them. Then we demonstrate how one…

In this paper, the elliptic curves theory is used for solving the Diophantine equations $\sum_{i=1}^n a_ix_{i} ^6+\sum_{i=1}^m b_iy_{i} ^3= \sum_{i=1}^na_iX_{i}^6\pm\sum_{i=1}^m b_iY_{i} ^3$, where $n$, $m$ $\geq 1$ and $a_i$, $b_i$, are…

This brief survey deals with multi-dimensional Diophantine approximations in sense of linear form and with simultaneous Diophantine approximations. We discuss the phenomenon of degenerate dimension of linear subspaces generated by the best…

We show how sums of some $5th$ powers can be written as sums of some cubics

We provide the analytic forms of the distributions for the sum of ordered spacings. We do this both for the case where the boundaries are included in the calculation of the spacings and the case where they are excluded. Both the probability…

We establish arithmetical properties and provide essential bounds for bi-sequences of approximation coefficients associated with the natural extension of maps, leading to continued fraction-like expansions. These maps are realized as the…

We demonstrate how connections between graph theory and Diophantine approximation can be used in conjunction to give simple and accessible proofs of seemingly difficult results in both subjects.

We prove a new lower bound for the exponent of growth of the best two-dimensional Diophantine approximations with respect to Euclidean norm.

We obtain lower bound for the maximum distance between any three distinct points in an affine lattice which are close to a helix with small curvature and torsion.

We consider planar curved strictly convex domains with no or very weak smoothness assumptions and prove sharp bounds for square-functions associated to the lattice point discrepancy.

In a previous paper, we studied certain sequences of simultaneous rational approximations in ${\bf R}^2$ which present some analogy with the continued fractions. We got results around the Littlewood conjecture by using such approximations.…

We obtain asymptotic formulae with optimal error terms for the number of lattice points under and near a dilation of the standard parabola, the former improving upon an old result of Popov. These results can be regarded as achieving the…

I present here the proofs of results, which are obtained in my papers "On the linear forms with algebraic coefficoients of logarithms of algebraic numbers", VINITI, 1996, 1617-B96, pp. 1 - 23 (in Russian), and "On the systems of linear…