Related papers: Diophantine Approximation in local function fields…
In the present article, the author uses Fourier theory of tempered distributions (generalized functions) in deriving a formula for Dirichlet-like integrals. The applied method is remarkably efficient and allows a solution in a few…
This chapter presents controlled approximations of Kohn-Sham density functional theory (DFT) that enable very large scale simulations. The work is motivated by the study of defects in crystalline solids, though the ideas can be used in…
Simultaneous Diophantine approximation is concerned with the approximation of a point $\mathbf x\in\mathbb R^d$ by points $\mathbf r\in\mathbb Q^d$, with a view towards jointly minimizing the quantities $\|\mathbf x - \mathbf r\|$ and…
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…
The theory of uniform Diophantine approximation concerns the study of Dirichlet improvable numbers and the metrical aspect of this theory leads to the study of the product of consecutive partial quotients in continued fractions. It is known…
We determine the generic multiplicative approximation rate on a hypersurface. There are four regimes, according to convergence or divergence and curved or flat, and we address all of them. Using geometry and arithmetic in Fourier space, we…
We consider the homogeneous Dirichlet problem for the integral fractional Laplacian $(-\Delta)^s$. We prove optimal Sobolev regularity estimates in Lipschitz domains provided the solution is $C^s$ up to the boundary. We present the…
Following the work of Waldschmidt, we investigate problems in Diophantine approximation on abelian varieties. First we show that a conjecture of Waldschmidt for a given simple abelian variety is equivalent to a well-known Diophantine…
We generalize the Brezzi-Rappaz-Raviart approximation theorem, which allows to obtain existence and a priori error estimates for approximations of solutions to some nonlinear partial differential equations. Our contribution lies in the fact…
Schmidt generalized in 1967 the theory of classical Diophantine approximation to subspaces of $\R^n$. We consider Diophantine exponents for linear subspaces of $\R^n$ which generalize the irrationality measure for real numbers. Using…
We show that an interesting class of functionals of stochastic differential equations can be approximated by a Chen-Fliess series of iterated stochastic integrals and give a L^{2} error estimate, thus generalizing the standard stochastic…
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…
Using the circle method in combination with lattice point counting arguments, we show that for almost all homogeneous diophantine equations of additive type and degree $k$ in more than $4k$ variables, the Local-Global principle holds true.…
In 2004, J.C. Tong found bounds for the approximation quality of a regular continued fraction convergent of a rational number, expressed in bounds for both the previous and next approximation. We sharpen his results with a geometric method…
In this extended abstract we deal with the relations between the numerical/diophantine approximation and the symbolic/algebraic geometry approachs to solving of multivariate diophentine polynomial systems, obtaining several consecuences…
We characterize the uniform limits of Dirichlet polynomials on a right half plane. In the Dirichlet setting, we find approximation results, with respect to the Euclidean distance and {to} the chordal one as well, analogous to classical…
We consider the problem of discretizing one-dimensional, real-valued functions as graphs. The goal is to find a small set of points, from which we can approximate the remaining function values. The method for approximating the unknown…
The inhomogeneous Khintchine-Groshev Theorem is a classical generalization of Khintchine's Theorem in Diophantine approximation, by approximating points in $\mathbb{R}^m$ by systems of linear forms in $n$ variables. Analogous to the…
In this paper, we study an analytic curve $\varphi: I=[a,b]\rightarrow \mathrm{M}(m\times n, \mathbb{R})$ in the space of $m$ by $n$ real matrices, and show that if $\varphi$ satisfies certain geometric condition, then for almost every…
Point counting estimates are a key stepping stone to various results in metric Diophantine approximation. In this paper we use the quantitative non-divergence estimates originally developed by Kleinbock and Margulis to improve lower bounds…