Related papers: Diophantine exponents of measures: a dynamical app…
In this note, we present an improvement to a recent result due to Beresnevich, Levesley, and Ward (2021) pertaining to weighted simultaneous Diophantine approximation on manifolds.
Given a compact pseudo-metric space, we associate to it upper and lower dimensions, depending only on the metric. Then we construct a doubling metric for which the measure of a dillated ball is closely related to these dimensions.
We draw connections between the various conjectures which are included in G. R\'emond's generalized Lehmer problems. Specifically, we show that the degree one form of his conjecture for the multiplicative group is, in a sense, almost as…
In this paper, we consider the intrinsic Diophantine approximation on the triadic Cantor set $\mathcal{K}$, i.e. approximating the points in $\mathcal{K}$ by rational numbers inside $\mathcal{K}$, a question posed by K. Mahler. By using…
We extend the classical theorems of Khintchine and Schmidt in metric Diophantine approximation to the context of self-similar measures on $\mathbb{R}^d$. For this, we establish effective equidistribution of associated random walks on…
Consider the sum of the first $N$ eigenspaces for the Laplacian on a Riemannian manifold. A basis for this space determines a map to Euclidean space and for $N$ sufficiently large the map is an embedding. In analogy with a fruitful idea of…
Schmidt generalized in 1967 the theory of classical Diophantine approximation to subspaces of $\R^n$. We consider Diophantine exponents for linear subspaces of $\R^n$ which generalize the irrationality measure for real numbers. Using…
We use the theory of arithmetic quotients of the Bruhat-Tits tree developed by Serre and others to obtain Dirichlet-style theorems for Diophantine approximation on global function fields. This approach allows us to find sharp values for the…
With a view to establishing measure theoretic approximation properties of Delone sets, we study a setup which arises naturally in the problem of averaging almost periodic functions along exponential sequences. In this setting, we establish…
In this article we establish two new results on quantitative Diophantine approximation for one-parameter families of diagonal ternary indefinite forms. In the first result, we consider quadratic forms taking values at prime points. In the…
A periodic lattice in Euclidean space is the infinite set of all integer linear combinations of basis vectors. Any lattice can be generated by infinitely many different bases. This ambiguity was only partially resolved, but standard…
The Hausdorff dimension of the set of simultaneously tau well approximable points lying on a curve defined by a polynomial P(X)+alpha, where P(X) is a polynomial with integer coefficients and alpha is in R, is studied when tau is larger…
We extend the Khintchine transference inequalities, as well as a homogeneous-inhomogeneous transference inequality for lattices, due to Bugeaud and Laurent, to a weighted setting. We also provide applications to inhomogeneous Diophantine…
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…
The use of Hausdorff measures and dimension in the theory of Diophantine approximation dates back to the 1920s with the theorems of Jarnik and Besicovitch regarding well-approximable and badly-approximable points. In this paper we consider…
Using the variational principle in parametric geometry of numbers, we compute the Hausdorff and packing dimension of Diophantine sets related to exponents of Diophantine approximation, and their intersections. In particular, we extend a…
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…
We discuss Mahler's work on Diophantine approximation and its applications to Diophantine equations, in particular Thue-Mahler equations, S-unit equations and S-integral points on elliptic curves, and go into later developments concerning…
In 1995, Hill and Velani introduced the shrinking targets theory. Given a dynamical system $([0,1],T)$, they investigated the Hausdorff dimension of sets of points whose orbits are close to some fixed point. In this paper, we study the sets…
The well known theorems of Khintchine and Jarn\'ik in metric Diophantine approximation provide comprehensive description of the measure theoretic properties of real numbers approximable by rational numbers with a given error. Various…