Related papers: A measure theoretic result for approximation by De…

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?…

In Diophantine approximation, inhomogeneous problems are linked with homogeneous ones by means of the so-called Transference Theorems. We revisit this classical topic by introducing new exponents of Diophantine approximation. We prove 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.

Gallagher's theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto,…

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…

We develop the metric theory of Diophantine approximation on homogeneous varieties of semisimple algebraic groups and prove results analogous to the classical Khinchin and Jarnik theorems. In full generality our results establish…

Measurable sets are defined as those locally approximable, in a certain sense, by sets in the given algebra (or ring). A corresponding measure extension theorem is proved. It is also shown that a set is locally approximable in the mentioned…

We place the theory of metric Diophantine approximation on manifolds into a broader context of studying Diophantine properties of points generic with respect to certain measures on $\Bbb R^n$. The correspondence between multidimensional…

We generalize Dirichlet's diophantine approximation theorem to approximating any real number $\alpha$ by a sum of two rational numbers $\frac{a_1}{q_1} + \frac{a_2}{q_2}$ with denominators $1 \leq q_1, q_2 \leq N$. This turns out to be…

In this paper we provide an approximation \`a la Ambrosio-Tortorelli of some classical minimization problems involving the length of an unknown one-dimensional set, with an additional connectedness constraint, in dimension two. We introduce…

Let $\al$ be an irrational and $\varphi: \N \rightarrow \R^+$ be a function decreasing to zero. For any $\al$ with a given Diophantine type, we show some sharp estimations for the Hausdorff dimension of the set [E_{\varphi}(\al):={y\in \R:…

We discuss a method to estimate the measure of a compact set which is approximated using the Hausdorff distance by a sequence of compact sets. We do this by considering corresponding fattenings of the sequence of compact sets and showing…

We prove that almost all real numbers (with respect to Lebesgue measure) are approximated by the convergents of their $\beta$-expansions with the exponential order $\beta^{-n}$. Moreover, the Hausdorff dimensions of sets of the real numbers…

Zero-one laws are a central topic in metric Diophantine approximation. A classical example of such laws is the Borel-Bernstein theorem. In this note, we prove a complex analogue of the Borel-Bernstein theorem for complex Hurwitz continued…

Let $(\Omega, \mathcal{A}, \mu)$ be a probability space. The classical Borel-Cantelli Lemma states that for any sequence of $\mu$-measurable sets $E_i$ ($i=1,2,3,\dots$), if the sum of their measures converges then the corresponding…

In this paper we develop a measure-theoretic method to treat problems in hypergraph theory. Our central theorem is a correspondence principle between three objects: An increasing hypergraph sequence, a measurable set in an ultraproduct…

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…

Fundamental questions in Diophantine approximation are related to the Hausdorff dimension of sets of the form $\{x\in \mathbb{R}: \delta_x = \delta\}$, where $\delta \geq 1$ and $\delta_x$ is the Diophantine approximation rate of an…

This brief survey deals with multi-dimensional Diophantine approximations in sense of linear form and with simultaneous Diophantine approximations. We discuss the phenomenon of degenerate dimension of linear subspaces generated by the best…

Function approximation is a generic process in a variety of computational problems, from data interpolation to the solution of differential equations and inverse problems. In this work, a unified approach for such techniques is…