Related papers: A zero-one law for improvements to Dirichlet's the…

In this paper we prove an upper bound on the "size" of the set of multiplicatively $\psi$-approximable points in $\mathbb R^d$ for $d>1$ in terms of $f$-dimensional Hausdorff measure. This upper bound exactly complements the known lower…

The main question studied in this article may be viewed as a nonlinear analogue of Dvoretzky's theorem in Banach space theory or as part of Ramsey theory in combinatorics. Given a finite metric space on n points, we seek its subspace of…

Minkowski's First Theorem and Dirichlet's Approximation Theorem provide upper bounds on certain minima taken over lattice points contained in domains of Euclidean spaces. We study the distribution of such minima and show, under some…

On the space $\mathcal{L}_{n+1}$ of unimodular lattices in $\mathbb{R}^{n+1}$, we consider the standard action of $a(t)=\mathrm{diag}(t^n,t^{-1},\ldots,t^{-1})\in \mathrm{SL}(n+1,\mathbb{R})$ for $t>1$. Let $M$ be a nondegenerate…

Khintchine's theorem on the measure dichotomy for the set of $\psi$-approximable numbers has been generalized to inhomogeneous and higher-dimensional settings. Allen and Ram\'irez conjectured that the monotonicity condition can be removed…

$ \newcommand{\schs}{\scriptstyle{\mathsf{S}}_1} $For all $n \ge 1$, we give an explicit construction of $m \times m$ matrices $A_1,\ldots,A_n$ with $m = 2^{\lfloor n/2 \rfloor}$ such that for any $d$ and $d \times d$ matrices…

Following the development of weighted asymptotic approximation properties of matrices, we introduce the analogous uniform approximation properties (that is, study the improvability of Dirichlet's Theorem). An added feature is the use of…

Given two elliptic operators L and M in nondivergence form, with coefficients A_L(x), A_M(x) and drift terms b_L(x), b_M(x), respectively, satisfying a Carleson measure disagreement condition in a Lipschitz domain Omega in R^{n+1}, then…

This work is motivated by a paper of Davenport and Schmidt, which treats the question of when Dirichlet's theorems on the rational approximation of one or of two irrationals can be improved and if so, by how much. We consider a…

Let $[a_1(x),a_2(x),\ldots, a_n(x), \ldots]$ be the continued fraction expansion of an irrational number $x\in (0, 1)$. The study of the growth rate of the product of consecutive partial quotients $a_n(x)a_{n+1}(x)$ is associated with the…

Given a nonnegative function $\psi : \N \to \R $, let $W(\psi)$ denote the set of real numbers $x$ such that $|nx -a| < \psi(n) $ for infinitely many reduced rationals $a/n (n>0) $. A consequence of our main result is that $W(\psi)$ is of…

Let $\nu$ be a finite complex measure with support in $\bar {\mathbb D}$ and let $\mathcal C\nu$ denote the Cauchy transform of $\nu .$ Suppose that $\nu$ annihilates polynomials in complex variable $z$ and $\nu |_{\partial \mathbb D} =…

Given a sequence of real numbers $\{\psi(n)\}_{n\in\mathbb{N}}$ with $0\leq \psi(n)<1$, let $W(\psi)$ denote the set of $x\in[0,1]$ for which $|xn-m|<\psi(n)$ for infinitely many coprime pairs $(n,m)\in\mathbb{N}\times\mathbb{Z}$. The…

The present paper establishes the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO anti-symmetric part. In…

We prove that the Dirichlet problem for degenerate elliptic equations $\mathrm{div}(A \nabla u) = 0$ in the upper half-space $(x,t)\in \mathbb{R}^{n+1}_+$ is solvable when $n\geq2$ and the boundary data is in $L^p_\mu(\mathbb{R}^n)$ for…

We develop the classical theory of Diophantine approximation without assuming monotonicity or convexity. A complete `multiplicative' zero-one law is established akin to the `simultaneous' zero-one laws of Cassels and Gallagher. As a…

We show that there exists a positive arithmetical formula $\psi(x,R)$, where $x \in \omega$, $R \subseteq \omega$, with no hyperarithmetical fixed point. This answers a question of Gerhard J\"{a}ger. As corollaries we obtain results on the…

In this article, we establish the Picard-Lindelof theorem and approximating results for dynamic equations on time scale. We present a simple proof for the existence and uniqueness of the solution. The proof is produced by using convergence…

We consider a quasilinear elliptic equation involving a first order term, under zero Dirichlet boundary condition in half spaces. We prove that any positive solution is monotone increasing w.r.t. the direction orthogonal to the boundary.…

We establish $L^p$ solvability of the Dirichlet problem, for some finite $p$, in a 1-sided chord-arc domain $\Omega$ (i.e., a uniform domain with Ahlfors-David regular boundary), for elliptic equations of the form \[ Lu=-\text{div}(A\nabla…