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The classical Khintchine-Groshev theorem is a generalization of Khintchine's theorem on simultaneous Diophantine approximation, from approximation of points in $\mathbb R^m$ to approximation of systems of linear forms in $\mathbb R^{nm}$.…

The classical Khintchine--Jarn\'ik Theorem provides elegant criteria for determining the Lebesgue measure and Hausdorff measure of sets of points approximated by rational points, which has inspired much modern research in metric Diophantine…

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…

We give a new Hausdorff content bound for limsup sets, which is related to Falconer's sets of large intersection. Falconer's sets of large intersection satisfy a content bound for all balls in a space. In comparison, our main theorem only…

This paper focuses on the metric properties of L\"uroth well approximable numbers, studying analogous of classical results, namely the Khintchine Theorem, the Jarn\'ik--Besicovitch Theorem, and the result of Dodson. A supplementary proof is…

The Duffin--Schaeffer Conjecture answers a question on how well one can approximate irrationals by rational numbers in reduced form (an imposed condition) where the accuracy of the approximation depends on the rational number. It can be…

Metric Diophantine approximation in its classical form is the study of how well almost all real numbers can be approximated by rationals. There is a long history of results which give partial answers to this problem, but there are still…

We prove a version of the Khinchine--Groshev theorem for Diophantine approximation of matrices subject to a congruence condition. The proof relies on an extension of the Dani correspondence to the quotient by a congruence subgroup. This…

Khintchine's theorem is a classical result from metric number theory which relates the Lebesgue measure of certain limsup sets with the convergence/divergence of naturally occurring volume sums. In this paper we ask whether an analogous…

In this paper we improve estimates of Jarnik and Apfelbeck for uniform Diophantine exponents of transposed systems of linear forms and generalize to the case of an arbitrary system the estimates of Laurent and Bugeaud for individual…

Given $n\in\mathbb{N}$ and $\tau>\frac1n$, let $\mathcal{S}_n(\tau)$ denote the classical set of $\tau$-approximable points in $\mathbb{R}^n$, which consists of ${\bf x}\in \mathbb{R}^n$ that lie within distance $q^{-\tau-1}$ from the…

We prove a multidimensional weighted analogue of the well-known theorem of Kurzweil (1955) in the metric theory of inhomogeneous Diophantine approximation. Let $A$ be matrix of real numbers, $\Psi$ an $n$-tuple of monotonic decreasing…

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 paper we initiate a new approach to studying approximations by rational points to points on smooth submanifolds of $\mathbb{R}^n$. Our main result is a convergence Khintchine type theorem for arbitrary nondegenerate submanifolds of…

The Khintchine-Groshev theorem in Diophantine approximation theory says that there is a dichotomy of the Lebesgue measure of sets of $\psi$-approximable numbers, given a monotonic function $\psi$. Allen and Ram\'irez removed the…

The main objective of this paper is to prove a Khintchine type theorem for divergence for linear Diophantine approximation on non-degenerate manifolds, which completes earlier results for convergence.

In 1926 Khintchine introduced a topological argument proving the existence of uncountably many nontrivial singular linear forms of $n \geq 2$ variables. Throughout the years, this argument has been extensively modified and generalized. Most…

This paper is motivated by recent applications of Diophantine approximation in electronics, in particular, in the rapidly developing area of Interference Alignment. Some remarkable advances in this area give substantial credit to the…

We determine the generic multiplicative approximation rate on a hypersurface. There are four regimes, according to convergence or divergence and curved or flat, and we address all of them. Using geometry and arithmetic in Fourier space, we…

A general form of the Borel-Cantelli Lemma and its connection with the proof of Khintchine's Theorem on Diophantine approximation and the more general Khintchine-Groshev theorem are discussed. The torus geometry in the planar case allows a…