A simple proof of Schmidt-Summerer's inequality
In this paper we give a simple proof of an inequality for intermediate Diophantine exponents obtained recently by W. M. Schmidt and L. Summerer.
math.NT↗
arXiv subjects
Oleg N. German
Publications and source records attributed to Oleg N. German.
Intermediate Diophantine exponents and parametric geometry of numbers
This is a revised compilation of the papers arXiv:1105.1554 and arXiv:1105.5823. We develop some of the ideas belonging to W.Schmidt and L.Summerer to define intermediate Diophantine exponents and split several transference inequalities into a chain of inequalities for intermediate exponents.
math.NT↗
Splitting transference inequalities with the help of Wolfgang Schmidt's parametric geometry of numbers
This paper is a sequel to our previous paper arXiv:1105.1554, where we defined two types of intermediate Diophantine exponents, connected them to Schmidt exponents and split Dyson's transference inequality into a chain of inequalities for intermediate exponents. Here we present splitting of some other transference inequalities involving both regular and uniform Diophantine exponents.
math.NT↗
How to split Dyson's transference inequality with the help of Wolfgang Schmidt's parametric geometry of numbers
In this paper we develop some of the ideas belonging to W.Schmidt and L.Summerer to define intermediate Diophantine exponents and split Dyson's transference inequality into a chain of inequalities for intermediate exponents. This splitting generalizes the analogous result of M.Laurent and Y.Bugeaud for Khintchine's transference inequalities.
math.NT↗
Transference inequalities for multiplicative Diophantine exponents
In this paper we prove inequalities for multiplicative analogues of Diophantine exponents, similar to the ones known in the classical case. Particularly, we show that a matrix is badly approximable if and only if its transpose is badly approximable and establish some inequalities connecting multiplicative exponents with ordinary ones.
math.NT↗
An inequality for Diophantine exponents in the case of simultaneous approximation
In this paper we prove an inequality for individual and uniform Diophantine exponents in the case of simultaneous approximation. This inequality is better than Jarnik's for small values of the uniform exponent.
math.NT↗
On Diophantine exponents and Khintchine's transference principle
In this paper we improve estimates of Jarnik and Apfelbeck for uniform Diophantine exponents of transposed systems of linear forms and generalize to the case of an arbitrary system the estimates of Laurent and Bugeaud for individual exponents. The method proposed also gives a better constant in Mahler's transference theorem.
math.NT↗
Linear forms of a given Diophantine type
We prove a result on the existence of linear forms of a given Diophantine type.
math.NT↗
Theorem on best Diophantine approximations for linear forms
We prove a new quantitative result on the degeneracy of the dimension of the subspace spanned by the best Diophantine approximations for a linear form.
math.NT↗
On multidimensional generalization of the Lagrange theorem on continued fractions
The Lagrange theorem on continued fractions states that a number is a quadratic surd if and only if its continued fraction expansion is eventually periodic. The current paper is devoted to a multidimensional generalization of this fact. As a multidimensional analog of continued fractions so called Klein polyhedra are considered: given an irrational lattice in an n-dimensional Euclidean space, its Klein polyhedron is defined as the convex hull of nonzero lattice points with nonnegative coordinates.
math.NT↗
Klein polyhedra and lattices with positive norm minima
A Klein polyhedron is defined as the convex hull of nonzero lattice points inside an orthant of $\R^n$. It generalizes the concept of continued fraction. In this paper facets and edge stars of vertices of a Klein polyhedron are considered as multidimensional analogs of partial quotients and quantitative characteristics of these ``partial quotients'', so called determinants, are defined. It is proved that the facets of all the $2^n$ Klein polyhedra generated by a lattice $\La$ have uniformly bounded determinants if and only if the facets and the edge stars of the vertices of the Klein polyhedron generated by $\La$ and related to the positive orthant have uniformly bounded determinants.
math.NT↗