Related papers: On transfer inequalities in Diophantine approximat…
We consider Khintchine type inequalities on the $p$-th moments of vectors of $N$ pairwise independent Rademacher random variables. We establish that an analogue of Khintchine's inequality cannot hold in this setting with a constant that is…
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 classical Khintchine and Jarn\'ik theorems, generalizations of a consequence of Dirichlet's theorem, are fundamental results in the theory of Diophantine approximation. These theorems are concerned with the size of the set of real…
We show that the parabola is of strong Khintchine type for convergence, which is the first result of its kind for curves. Moreover, Jarnik type theorems are established in both the simultaneous and the dual settings, without monotonicity on…
Fix an integer $n\ge 2$. To each non-zero point $\mathbf{u}$ in $\mathbb{R}^n$, one attaches several numbers called exponents of Diophantine approximation. However, as Khintchine first observed, these numbers are not independent of each…
We prove a quantitative theorem for Diophantine approximation by rational points on spheres. Our results are valid for arbitrary unimodular lattices and we further prove 'spiraling' results for the direction of approximates. These results…
We prove a Khintchine result for convergence of a multiplicative Diophantine set with restricted denominators on an arbitrary non-degenerate line. Specifically, given sequences of real numbers $\{a_n\}_{n\in\mathbb{N}},\,…
Sz{\"u}sz's inhomogeneous version (1958) of Khintchine's theorem (1924) gives conditions on $\psi:\mathbb{N}\to\mathbb{R}_{\geq 0}$ under which for almost every real number $\alpha$ there exist infinitely many rationals $p/q$ such that…
Analogues of the classical theorems of Khintchine, Jarnik and Jarnik-Besicovitch in the metrical theory of Diophantine approximation are established for quaternions by applying results on the measure of general `lim sup' sets.
We provide several results on the diophantine properties of continued fractions on the Heisenberg group, many of which are analogous to classical results for real continued fractions. In particular, we show an analog of Khinchin's theorem…
We establish several optimal moment comparison inequalities (Khinchin-type inequalities) for weighted sums of independent identically distributed symmetric discrete random variables which are uniform on sets of consecutive integers.…
We prove an analogue the Khinchin theorem for the Diophantine approximation by integer vectors lying on a quadratic variety. The proof is based on the study of a dynamical system on a homogeneous space of the orthogonal group. We show that…
There are two fundamental results in the classical theory of metric Diophantine approximation: Khintchine's theorem and Jarnik's theorem. The former relates the size of the set of well approximable numbers, expressed in terms of Lebesgue…
This article investigates sharp comparison of moments for various classes of random variables appearing in a geometric context. In the first part of our work we find the optimal constants in the Khintchine inequality for random vectors…
We prove an easy statement about inhomogeneous approximation in metric theory of Diophantine Approximation.
We give a strengthening of the classical Khintchine inequality between the second and the $p$-th moment for $p \ge 3$ with optimal constant by adding a deficit depending on the vector of coefficients of the Rademacher sum.
The present paper is concerned with equidistribution results for certain flows on homogeneous spaces and related questions in Diophantine approximation. Firstly, we answer in the affirmative, a question raised by Kleinbock, Shi and Weiss…
In this paper we develop some of the ideas belonging to W.Schmidt and L.Summerer to define intermediate Diophantine exponents and split Dyson's transference inequality into a chain of inequalities for intermediate exponents. This splitting…
Recently, Ghosh \& Haynes \cite{HG} proved a Khintchine-type result for the problem of Diophantine approximation in certain projective spaces. In this note we complement their result by observing that a Jarn\'{\i}k-type result also holds…
In this paper we show that the subset of integers that satisfies the Khintchine inequality for $p=1$ with the optimal constant ${\sqrt{2}}$ has to be a $Z_2$ set. We further prove a similar result for a large class of discrete groups. Our…