Related papers: Restricted simultaneous Diophantine approximation

We study the problem of computing the value function from a discretely-observed trajectory of a continuous-time diffusion process. We develop a new class of algorithms based on easily implementable numerical schemes that are compatible with…

This paper develops a new framework, \emph{simultaneous saturation}, designed to quantify the size of sets whose elements are simultaneously large. The framework establishes a correspondence between the magnitude of such sets and a system…

The goal of this survey is to discuss the Quantitative non-Divergence estimate on the space of lattices and present a selection of its applications. The topics covered include extremal manifolds, Khintchine-Groshev type theorems, rational…

In a ground-breaking work \cite{BY}, Beresnevich and Yang recently proved Khintchine's theorem in simultaneous Diophantine approximation for nondegenerate manifolds resolving a long-standing problem in the theory of Diophantine…

Let $\Omega $ be a bounded domain in $\mathbb{R}^{d}$ $\left( d\geq 2\right) $ pretty regular. We solve the variational Dirichlet problem for a class of quasi-linear elliptic systems.

The method of constructing approximate solutions of the first boundary value problem for linear differential equations based on incomplete (even and odd) trigonometric splines is considered. The theoretical positions are illustrated by…

We study the optimal rectangular-discrepancy approximation of permutons by finite permutations. We transfer bounds from discrepancy theory to this more restricted setup. Moreover, we show that superlinear approximation can occur only for…

This article develops an approximate proximal approach for the generalized method of lines. The present results are extensions and applications of previous ones which have been published since 2011, in books and articles such as [3,4,5,6].…

In these notes, we begin by recalling aspects of the classical theory of metric Diophantine approximation; such as theorems of Khintchine, Jarn\'{\i}k, Duffin-Schaeffer and Gallagher. We then describe recent strengthening of various…

We establish a strong form of Littlewood's conjecture with inhomogeneous shifts, for a full-dimensional set of pairs of badly approximable numbers on a vertical line. We also prove a uniform assertion of this nature, generalising a strong…

We study the shrinking target problem for actions of semisimple groups on homogeneous spaces, with applications to logarithm laws and Diophantine approximation.

We establish several new inequalities linking classical exponents of Diophantine approximation associated to a real vector $\underline{\xi}=(\xi,\xi^{2},\ldots,\xi^{N})$, in various dimensions $N$. We thereby obtain variants, and partly…

We focus on the linear convergence of generalized proximal point algorithms for solving monotone inclusion problems. Under the assumption that the associated monotone operator is metrically subregular or that the inverse of the monotone…

In this paper we prove transference inequalities for regular and uniform Diophantine exponents in the weighted setting. Our results generalize the corresponding inequalities that exist in the `non-weighted' case.

In this paper, we study Diophantine exponents $w_n$ and $w_n ^{*}$ for Laurent series over a finite field. Especially, we deal with the case $n=2$, that is, quadratic approximation. We first show that the range of the function $w_2-w_2…

We present a proof of a multidimensional version of Peres-Schlag's theorem on Diophantine approximations with lacunary sequences.

The distribution of $\alpha p$ modulo one, where $p$ runs over the rational primes and $\alpha$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $\nu>0$ one can establish the infinitude…

We use a numerical-analytic technique to construct a sequence of successive approximations to the solution of a system of fractional differential equations, subject to Dirichlet boundary conditions. We prove the uniform convergence of the…

We present and study approximate notions of dimensional and margin complexity, which correspond to the minimal dimension or norm of an embedding required to approximate, rather then exactly represent, a given hypothesis class. We show that…

We introduce a relaxation for homomorphism problems that combines semidefinite programming with linear Diophantine equations, and propose a framework for the analysis of its power based on the spectral theory of association schemes. We use…