Related papers: Diophantine approximation exponents on homogeneous…
Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are…
The present paper establishes qunatitative estimates on the rate of diophantine approximation in homogeneous varieties of semisimple algebraic groups. The estimates established generalize and improve previous ones, and are sharp in a number…
Given any line in the plane, we strengthen the Littlewood conjecture by two logarithms for almost every point on the line, thereby generalising the fibre result of Beresnevich, Haynes, and Velani. To achieve this, we prove an effective…
We discuss the problem of finding optimal exponents in Diophantine estimates involving one real number and, in some cases where such an exponent is known, present some properties of the corresponding extremal numbers.
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.
The nature and origin of exceptional sets associated with the rotation number of circle maps, Kolmogorov-Arnol'd-Moser theory on the existence of invariant tori and the linearisation of complex diffeomorphisms are explained. The metrical…
In this paper we study the spectrum of weak uniform Diophantine exponents of lattices and obtain its complete description in the two-dimensional case.
We give an elementary proof of a recent metrical Diophantine result by D. Kleinbock related to badly approximable vectors in affine subspaces.
We improve on a result by Svetlana Jitomirskaya and Wencai Liu dealing with inhomogeneous Diophantine approximation in the coprime setting.
In this article, we characterize two kinds of exceptional orbits of the geodesic flow associated with the Modular surface in terms of a two-parameter family of continued fraction expansion of endpoints of the lifts to the hyperbolic plane…
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…
In this paper we study $p$-adic Diophantine approximation on manifolds, specifically multiplicative Diophantine approximation on affine subspaces and a Diophantine dichotomy for analytic $p$-adic manifolds.
A basic question of Diophantine approximation, which is the first issue we discuss, is to investigate the rational approximations to a single real number. Next, we consider the algebraic or polynomial approximations to a single complex…
In this paper we prove an existence theorem concerning linear forms of a given Diophantine type and apply it to study the structure of the spectrum of lattice exponents.
We discuss some easy statements dealing with linear inhomogeneous Diophantine approximation. Surprisingly, we did not find some of them in the literature.
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…
Let $\theta\in\mathbb{R}^d$. We associate three objects to each approximation $(p,q)\in \mathbb{Z}^d\times \mathbb{N}$ of $\theta$: the projection of the lattice $\mathbb{Z}^{d+1}$ to the hyperplane of the first $d$ coordinates along the…
We study the problem of best approximations of a vector $\alpha\in{\mathbb R}^n$ by rational vectors of a lattice $\Lambda\subset {\mathbb R}^n$ whose common denominator is bounded. To this end we introduce successive minima for a periodic…
We develop the theory of Diophantine approximation for systems of simultaneously small linear forms, which coefficients are drawn from any given analytic non-degenerate manifolds. This setup originates from a problem of Sprind\v{z}uk from…
We consider multiparameter dynamics on the space of unimolular lattices. Along with quantitative nondivergence we prove that multiplicative Diophantine exponents of hyperplanes are inherited by their nondegenerate submanifolds.