Related papers: Exponents of Diophantine approximation
The paper is devoted to the problem of estimating the constant of the best Diophantine approximations. The estimates of lower bound $ C_n $ for $ n = 5 $ and $ n = 6 $ was improved. The first chapter gives an overview of the history of…
We introduce an inhomogeneous variant of Kaufman's measure, with applications to diophantine approximation. In particular, we make progress towards a problem related to Littlewood's conjecture.
We associate certain curves over function fields to given algebraic power series and show that bounds on the rank of Kodaira-Spencer map of this curves imply bounds on the exponents of the power series, with more generic curves giving lower…
We give upper and lower bounds for Diophantine exponents measuring how well a point in the plane can be approximated by points in the orbit of a lattice $\Gamma<\mathrm{SL}_2(\mathbb{R})$ acting linearly on $\mathbb{R}^2$. Our method gives…
We elaborate on a problem raised by Schmidt in 1967 which generalizes the theory of classical Diophantine approximation to subspaces of $\R^n$. We consider Diophantine exponents for linear subspaces of $\R^n$ which generalize the…
This paper is devoted to the study of a problem of Cassels in multiplicative Diophantine approximation which involves minimising values of a product of affine linear forms computed at integral points. It was previously known that values of…
In this paper we give a simple proof of an inequality for intermediate Diophantine exponents obtained recently by W. M. Schmidt and L. Summerer.
The objective of this paper is to (partially) address the issue of finding an analogue to Khintchine's theorem for IFS Fractals. We study the convergence case for Diophantine approximations, and show an improved result for higher…
In this paper, we study Diophantine exponents $w_n$ and $w_n ^{*}$ for Laurent series over a finite field. Especially, we deal with the case $n=2$, that is, quadratic approximation. We first show that the range of the function $w_2-w_2…
This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable numbers $\zeta_{1},\zeta_{2},...,\zeta_{k}$. The approach relies on results on the connection between the set of all…
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 investigate the number of integer solutions to a multiplicative Diophantine approximation problem and show that the associated counting function converges in distribution to a normal law. Our approach relies on the analysis of…
We prove a theorem about approximation to an irrational number by rational numbers whose denominator n is free of prime factors bigger than a power of log n. We strengthen the result in version 1 by using an exponential sum over smooth…
In this paper we give a short overview about the Ball-Evans approximation problem, i.e. about the approximation of Sobolev homeomorphism by a sequence of diffeomorphisms (or piecewise affine homeomorphisms) and we recall the motivation for…
In this paper, we consider the problem of counting Diophantine inequalities with multiple natural constraints. We prove a very general result in this setting using dynamical techniques. More precisely, we consider the joint asymptotic…
Using the Parametric Geometry of Numbers introduced recently by W.M. Schmidt and L. Summerer and results by D. Roy, we establish that the spectrum of the $2n$ exponents of Diophantine approximation in dimension $n\geq3$ is a subset of…
We study properties of Diophantine exponents of lattices and so-called related "weak" uniform approximations introduced in recent papers by Oleg German, in the simplest two-dimensional case. In contrast to the multidimensional case, in the…
For any given positive definite binary quadratic form $Q$ with integer coefficients, we establish two results on Diophantine approximation with integers represented by $Q$. Firstly, we show that for every irrational number $\alpha$, there…
In this paper, we explicitly find all solutions of the title Diophantine equation, using lower bounds for linear forms in logarithms and properties of continued fractions. Further, we use a version of the Baker-Davenport reduction method in…
We develop the metric theory of Diophantine approximation on homogeneous varieties of semisimple algebraic groups and prove results analogous to the classical Khinchin and Jarnik theorems. In full generality our results establish…