Related papers: Expanding translates of curves and Dirichlet-Minko…
We study the Lorentzian Calder\'on problem, where the objective is to determine a globally hyperbolic Lorentzian metric up to a boundary fixing diffeomorphism from boundary measurements given by the hyperbolic Dirichlet-to-Neumann map. This…
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 prove a sharp analogue of Minkowski's inhomogeneous approximation theorem over fields of power series $\mathbb{F}_q((T^{-1}))$. Furthermore, we study the approximation to a given point $\underline{y}$ in $\mathbb{F}_q((T^{-1}))^2$ by the…
Given a symmetric convex body $C$ and $n$ hyperplanes in an Euclidean space, there is a translate of a multiple of $C$, at least ${1\over n+1}$ times as large, inside $C$, whose interior does not meet any of the hyperplanes. The result…
A basic question of Diophantine approximation, which is the first issue we discuss, is to investigate the rational approximations to a single real number. Next, we consider the algebraic or polynomial approximations to a single complex…
In this article, we derive the asymptotic expansion, up to an arbitrary order in theory, for the solution of a two-dimensional elliptic equation with strongly anisotropic diffusion coefficients along different directions, subject to 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…
The goal of this paper is to develop the theory of weighted Diophantine approximation of rational numbers to $p$-adic numbers. Firstly, we establish complete analogues of Khintchine's theorem, the Duffin-Schaeffer theorem and the…
We establish that the Dirichlet problem for convex linear growth functionals on $BD$, the functions of bounded deformation, gives rise to the same unconditional Sobolev and partial $C^{1,\alpha}$-regularity theory as presently available for…
We study expansions near the boundary of solutions to the Dirichlet problem for the constant mean curvature equation in the hyperbolic space. With a characterization of remainders of the expansion by multiple integrals, we establish optimal…
The Hurwitz chain gives a sequence of pairs of Farey approximations to an irrational real number. Minkowski gave a criterion for a number to be algebraic by using a certain generalization of the Hurwitz chain. We apply Minkowski's…
We give a fast, exact algorithm for solving Dirichlet problems with polynomial boundary functions on quadratic surfaces in R^n such as ellipsoids, elliptic cylinders, and paraboloids. To produce this algorithm, first we show that every…
The Dirichlet problem for a Monge-Ampere equation corresponding to a nonnegative, possible degenerate cohomology class on a Kaehler manifold with boundary is studied. C^{1,\alpha} estimates away from a divisor are obtained, by combining…
We study the Folklore set of Dirichlet improvable matrices in $\mathbb R^{m\times n}$ which are neither singular nor badly approximable. We prove the non-emptiness for all positive integer pairs $m,n$ apart from $\{m,n\}=\{ 1,1\}$ and…
A large class of quantum field theories on 1+1 dimensional Minkowski space, namely, certain integrable models, has recently been constructed rigorously by Lechner. However, the construction is very abstract and the concrete form of local…
We consider the Dirichlet problem for a class of elliptic and parabolic equations in the upper-half space $\mathbb{R}^d_+$, where the coefficients are the product of $x_d^\alpha, \alpha \in (-\infty, 1),$ and a bounded uniformly elliptic…
In this paper we study elliptic curves which have a number of points whose coordinates are in arithmetic progression. We first motivate this diophantine problem, prove some results, provide a number of interesting examples and, finally…
In this paper, we firstly introduce the group of similarity transformations in the Minkowski-3 space. We describe differential- geometric invariants of a non-lightlike curve according to the group of similarity transformations of the…
In twisted Diophantine approximation, for a fixed $m\times n$ matrix $\boldsymbol\alpha$ one is interested in sets of vectors $\boldsymbol\beta\in\mathbb R^m$ such that the system of affine forms $\mathbb R^n \ni \mathbf q \mapsto…
Let $\cal C$ be a non--degenerate planar curve and for a real, positive decreasing function $\psi$ let $\cal C(\psi)$ denote the set of simultaneously $\psi$--approximable points lying on $\cal C$. We show that $\cal C$ is of Khintchine…