author={Oleg German},
  • O. German
  • Published 4 May 2019
  • Mathematics
  • Mathematika
In this paper we prove transference inequalities for regular and uniform Diophantine exponents in the weighted setting. Our results generalize the corresponding inequalities that exist in the `non-weighted' case. 

Figures from this paper

Multiparametric geometry of numbers and its application to splitting transference theorems
  • O. German
  • Mathematics
    Monatshefte für Mathematik
  • 2022
In this paper we consider a multiparametric version of Wolfgang Schmidt and Leonard Summerer's parametric geometry of numbers. We apply this approach in two settings: the first one concerns weighted
Singular vectors on manifolds and fractals
We generalize Khintchine's method of constructing totally irrational singular vectors and linear forms. The main result of the paper shows existence of totally irrational vectors and linear forms
A graph arising in the Geometry of Numbers
The parametric geometry of numbers has allowed to visualize the simultaneous approximation properties of a collection of real numbers through the combined graph of the related successive minima
On Hausdorff dimension in inhomogeneous Diophantine approximation over global function fields
In this paper, we study inhomogeneous Diophantine approximation over the completion Kv of a global function field K (over a finite field) for a discrete valuation v, with affine algebra Rv. We obtain
Dimension estimates for badly approximable affine forms
For given ǫ ą 0 and b P R, we say that a real mˆn matrix A is ǫ-badly approximable for the target b if lim inf qPZn,}q}Ñ8 }q}xAq ́ by ě ǫ, where x ̈y denotes the distance from the nearest integral
Hausdorff measure of sets of Dirichlet non-improvable affine forms
Einige Bemerkungen \"uber inhomogene diophantische Approximationen
Es ist bekannt, dass für jedes reelle irrationale θ und für jedes reelle α unendlich viele Gitterpunkte (x, y) ∈ Z, x > 1 mit |θx− α− y| 6 1 x existieren. Es gibt viele klassische mehrdimensionale
A geometric proof of Jarnik's identity in the setting of weighted simultaneous approximation
Jarnik's identity plays a major role in classical simultaneous approximation to two real numbers. O. German [2] has shown a generalization to the weighted setting in which the identity has to be


On Diophantine transference principles
Abstract We provide an extension of the transference results of Beresnevich and Velani connecting homogeneous and inhomogeneous Diophantine approximation on manifolds and provide bounds for
Diophantine transference inequalities: weighted, inhomogeneous, and intermediate exponents
We extend the Khintchine transference inequalities, as well as a homogeneous-inhomogeneous transference inequality for lattices, due to Bugeaud and Laurent, to a weighted setting. We also provide
There is no analogue to Jarník’s relation for twisted Diophantine approximation
Jarník gave a relation between the two most classical uniform exponents of Diophantine approximation in dimension 2. In this paper we consider a twisted case, between the classical and the
A strengthening of Mahler's transference theorem
We obtain new transference theorems that improve some classical theorems of Mahler. Our results are stated in terms of consecutive minima of pseudo-compound parallelepipeds.
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
Diophantine approximation in ⁿ
The diophantine approximation deals with the approximation of real numbers (or real vectors) with rational numbers (or rational vectors). See the article Wikipedia article Diophantine_approximation
Diophantine Approximations and Diophantine Equations
Siegel's lemma and heights.- Diophantine approximation.- The thue equation.- S-unit equations and hyperelliptic equations.- Diophantine equations in more than two variables.
On Exponents of Homogeneous and Inhomogeneous Diophantine Approximation
In Diophantine Approximation, inhomogeneous problems are linked with homogeneous ones by means of the so-called Transference Theorems. We revisit this classical topic by introducing new exponents of
An inhomogeneous transference principle and Diophantine approximation
In a landmark paper (‘Flows on homogeneous spaces and Diophantine approximation on manifolds’, Ann. of Math. (2) 148 (1998), 339–360.) Kleinbock and Margulis established the fundamental
Introduction to Diophantine Approximation
This article formalizes some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals and proves that the inequality |xθ − y| ≤ 1/x has infinitely many solutions by continued fractions.