Related papers: A quantitative Khintchine-Groshev theorem for S-ar…
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 a paper from 2010, Budarina, Dickinson and Levesley studied the rational approximation properties of curves parametrized by polynomials with integral coefficients in Euclidean space of arbitrary dimension. Assuming the dimension is at…
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…
We show that affine coordinate subspaces of dimension at least two in Euclidean space are of Khintchine type for divergence. For affine coordinate subspaces of dimension one, we prove a result which depends on the dual Diophantine type of…
The continuous version of a fundamental result of Khinchin says that a half-infinite torus line in the unit square $[0,1]^2$ exhibits superdensity, which is a best form of time-quantitative density, if and only if the slope of the geodesic…
We first recall the connection, going back to A. Thue, between rational approximation to algebraic numbers and integer solutions of some Diophantine equations. Next we recall the equivalence between several finiteness results on various…
Our main result concerns a perturbation of a classic theorem of Khintchine in Diophantine approximation. We give sufficient conditions on a sequence of positive real numbers $(\psi_n)_{n \in \mathbb{N}}$ and differentiable functions…
We refine Khintchine Transference Principle which relates the measure of simultaneous rational approximation of an $n$ real numbers with the measure of linear independence of these $n$ numbers. Khintchine's inequalities are known to be…
Fix $d\in\mathbb N$, and let $S\subseteq\mathbb R^d$ be either a real-analytic manifold or the limit set of an iterated function system (for example, $S$ could be the Cantor set or the von Koch snowflake). An $extrinsic$ Diophantine…
The convergence theory for the set of simultaneously $\psi$-approximable points lying on a planar curve is established. Our results complement the divergence theory developed in `Diophantine approximation on planar curves and the…
There are two main interrelated goals of this paper. Firstly we investigate the sums \[ S_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{n\|n\alpha-\gamma\|}~~~\text{and}~~~ R_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{\|n\alpha-\gamma\|}\,, \] where…
Many results related to quantitative problems in the metric theory of Diophantine approximation are asymptotic, such as the number of rational solutions to certain inequalities grows with the same rate almost everywhere modulo an asymptotic…
his paper deals with approximating properties of the newly defined $q$-generalization of the Sz\'{a}sz operators in the case $q>1$. Quantitative estimates of the convergence in the polynomial weighted spaces and the Voronovskaja's theorem…
It was discovered that there is a formal analogy between Nevanlinna theory and Diophantine approximation. Via Vojta's dictionary, the Second Main Theorem in Nevanlinna theory corresponds to Schmidt's Subspace Theorem in Diophantine…
In metric Diophantine approximation there are two main types of approximations: simultaneous and dual for both homogeneous and inhomogeneous settings. The well known measure-theoretic theorems of Khintchine and Jarn\'ik are fundamental in…
We present a novel proof of the Duffin-Schaeffer conjecture in metric Diophantine approximation. Our proof is heavily motivated by the ideas of Koukoulopoulos-Maynard's breakthrough first argument, but simplifies and strengthens several…
In this paper we develop the continuous averaging method of Treschev to work on the simultaneous Diophantine approximation and apply the result to give a new proof of the Nekhoroshev theorem. We obtain a sharp normal form theorem and an…
Analogues of Khintchine's Theorem in simultaneous Diophantine approximation in the plane are proved with the classical height replaced by fairly general planar distance functions or equivalently star bodies. Khintchine's transference…
In this article, we study an analytic curve $\varphi: I=[a,b]\rightarrow \mathrm{M}(n\times n, \mathbb{R})$ in the space of $n$ by $n$ real matrices, and show that if $\varphi$ satisfies certain geometric conditions, then for almost every…
In this paper we construct a new family of sets based on Diophantine approximation in the Euclidean space, and consider their applications in several problems in harmonic analysis. Our first application is on the Hausdorff dimension of our…