Related papers: On intrinsic and extrinsic rational approximation …
In this article, we explore the notion of infinity by studying Cantor's contribution to this field. A brief history of set theory is given. As an example of infinity, we consider Hilbert's famous hotel. A graphical construction is used to…
We consider the problem of approaching real numbers with rational numbers with prime denominator and with a single numerator allowed for each denominator. We then present a simple application, related to possible correlations between trace…
We compute the Hausdorff dimension of sets of very well approximable vectors on rational quadrics. We use ubiquitous systems and the geometry of locally symmetric spaces. As a byproduct we obtain the Hausdorff dimension of the set of rays…
In this paper, we investigate the fractal uncertainty principle (FUP) for discrete Cantor sets, which are determined by an alphabet from a base of digits. Consider the base of M digits and the alphabets of cardinality A such that all the…
The approximation of natural numbers subsets has always been one of the fundamental issues in computability theory. Computable approximation, $\Delta_2$-approximation, as well as introducing the generically computable sets have been some…
In this paper we study the derived sets for the rational deformations of multiple zeta-star values. By using the theory of bounded variation functions, we will give function decompositions which describe the metric structure of the derived…
We study a variant of the Falconer distance problem for dot products. In particular, for fractal subsets $A\subset \mathbb{R}^n$ and $a,x\in \mathbb{R}^n$, we study sets of the form \[ \Pi_x^a(A) := \{\alpha \in \mathbb{R} : (a-x)\cdot y=…
The article is devoted to the alternating Cantor series. It is proved that any real number belonging to $[a_0-1;a_0]$, where $a_0=\sum^{\infty} _{k=1} {\frac{d_{2k}-1}{d_1d_2...d_{2k}}} $, has no more than two representations by the series…
In this article we establish an asymptotic formula for the number of rational points, with bounded denominators, within a given distance to a compact submanifold $\mathcal{M}$ of $\mathbb{R}^M$ with a certain curvature condition. Our result…
Consider the vanishing locus of a real analytic function on $\mathbb{R}^n$ restricted to $[0,1]^n$. We bound the number of rational points of bounded height that approximate this set very well. Our result is formulated and proved in the…
We prove two results on Fractal Uncertainty Principle (FUP) for discrete Cantor sets with large alphabets. First, we give an example of an alphabet with dimension $\delta \in (\frac12,1)$ where the FUP exponent is exponentially small as the…
This paper uses W. M. Schmidt's idea formulated in 1967 to generalise the classical theory of Diophantine approximation to subspaces of $\mathbb{R}^n$. Given two subspaces of $\mathbb{R}^n$ $A$ and $B$ of respective dimensions $d$ and $e$…
Given two coprime integers $p\ge 2$ and $q \ge 3$, let $D_p\subset[0,1)$ consist of all rational numbers which have a finite $p$-ary expansion, and let $$ K(q, \mathcal{A})=\bigg\{ \sum_{i=1}^\infty \frac{d_i}{q^i}: d_i\in \mathcal{A}~…
This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarnik type theorems for submanifolds of R^n. These problems have attracted a lot of interest since Kleinbock and…
We give another proof of a result of Adamczewski and Bell concerning Mahler equations: A formal power series satisfying a $p-$ and a $q-$Mahler equation over ${\mathbb C}(x)$ with multiplicatively independent positive integers $p$ and $q$…
We analyze the singularities of rational inner functions on the unit bidisk and study both when these functions belong to Dirichlet-type spaces and when their partial derivatives belong to Hardy spaces. We characterize derivative…
By closely following a construction by Ganelius, we construct Faber rational functions that allow us to derive tight and explicit bounds on Zolotarev numbers. We use our results to bound the singular values of matrices, including…
It is well known that in dimension one the set of Dirichlet improvable real numbers consists precisely of badly approximable and singular numbers. We show that in higher dimensions this is not the case by proving that there exist continuum…
This paper is an exposition, with some new applications, of our results on the growth of entropy of convolutions. We explain the main result on $\mathbb{R}$, and derive, via a linearization argument, an analogous result for the action of…
Kingman's Theorem on skeleton limits---passing from limits as $n\to \infty $ along $nh$ ($n\in \mathbb{N}$) for enough $h>0$ to limits as $t\to \infty $ for $t\in \mathbb{R}$---is generalized to a Baire/measurable setting via a topological…