相关论文: Metric Diophantine approximation: The Khintchine--…
Recently, Beresnevich, Vaughan, Velani, and Zorin (arXiv: 1506.09049) gave some sufficient conditions for a manifold to be of Khinchin type for convergence. We show that their techniques can be used in a more optimal way to yield stronger…
We outline a proof of an analogue of Khintchine's Theorem in R^2, where the ordinary height is replaced by a distance function satisfying an irrationality condition as well as certain decay and symmetry conditions.
The Generalised Baker-Schmidt Problem (1970) concerns the Hausdorff measure of the set of $\psi$-approximable points on a nondegenerate manifold. Beresnevich-Dickinson-Velani (in 2006, for the homogeneous setting) and…
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…
Let $\cS_n(\psi_1,...,\psi_n)$ denote the set of simultaneously $(\psi_1,...,\psi_n)$--approximable points in $\R^n$ and $\cSM_n(\psi)$ denote the set of multiplicatively $\psi$--approximable points in $\R^n$. Let $\cM$ be a manifold in…
The paper is mostly a survey on recent results in Diophantine approximation, with emphasis on properties of exponents measuring various notions of Diophantine <approximation.
We investigate the question of how well points on a nondegenerate $k$-dimensional submanifold $M \subseteq \mathbb R^d$ can be approximated by rationals also lying on $M$, establishing an upper bound on the "intrinsic Dirichlet exponent"…
In this paper we discuss a general problem on metrical Diophantine approximation associated with a system of linear forms. The main result is a zero-one law that extends one-dimensional results of Cassels and Gallagher. The paper contains a…
A paradigm for a global algebraic number theory of the reals is formulated with the purpose of providing a unified setting for algebraic and transcendental number theory. This is achieved through the study of subgroups of nonstandard models…
Theorems of Khintchine, Groshev, Jarn\'ik, and Besicovitch in Diophantine approximation are fundamental results on the metric properties of $\Psi$-well approximable sets. These foundational results have since been generalised to the…
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…
In this chapter we introduce the theory of Diophantine approximation via a series of basic examples from information theory relevant to wireless communications. In particular, we discuss Dirichlet's theorem, badly approximable points,…
In this paper we study random iterated function systems. Our main result gives sufficient conditions for an analogue of a well known theorem due to Khintchine from Diophantine approximation to hold almost surely for stochastically…
Let $F \subseteq [0,1]$ be a set that supports a probability measure $\mu$ with the property that $ |\widehat{\mu}(t)| \ll (\log |t|)^{-A}$ for some constant $ A > 0 $. Let $\mathcal{A}= (q_n)_{n\in \mathbb{N}} $ be a sequence of natural…
Diophantine approximation is the problem of approximating a real number by rational numbers. We propose a version of this in which the numerators are approximately related to the denominators by a Laurent polynomial. Our definition is…
The primary goal of this paper is to complete the theory of metric Diophantine approximation initially developed in [Ann. of Math.(2) 166 (2007), p.367-426] for $C^3$ non-degenerate planar curves. With this goal in mind, here for the first…
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…
This paper develops the metric theory of simultaneous inhomogeneous Diophantine approximation on a planar curve with respect to multiple approximating functions. Our results naturally generalize the homogeneous Lebesgue measure and Hausdor?…
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 with large uniform Diophantine exponents on…
In 1962, Gallagher proved an higher dimensional version of Khintchine's theorem on Diophantine approximation. Gallagher's theorem states that for any non-increasing approximation function $\psi:\mathbb{N}\to (0,1/2)$ with…