A higher-dimensional Kurzweil theorem for formal Laurent series over finite fields

In a recent paper, Kim and Nakada proved an analogue of Kurzweil?s theorem for inhomogeneous Diophantine approximation of formal Laurent series over finite fields. Their proof used continued fraction theory and thus cannot be easily extended to simultaneous Diophantine approximation. In this note, we give another proof which works for simultaneous Diophantine approximation as well.     

Rational approximation to algebraic varieties and a new exponent of simultaneous approximation

This paper deals with two main topics related to Diophantine approximation. Firstly, we show that if a point on an algebraic variety is approximable by rational vectors to a sufficiently large degree, the approximating vectors must lie in the topological closure of the rational points on the variety. In many interesting cases, in particular if the set of rational points on the variety is finite, this closure does not exceed the set of rational points on the variety itself. This result enables easier proofs of several known results as special cases. The proof can be generalized in some way and encourages to define a new exponent of simultaneous approximation. The second part of the paper is devoted to the study of this exponent.     

Algebroid functions, Wirsing's theorem and their relations

In this paper, we first point out a relationship between the Second Main Theorem for algebriod functions in Nevanlinna theory and Wirsing's theorem in Diophantine approximation. This motivates a unified proof for both theorems. The second part of this paper deals with “moving targets” problem for holomorphic maps to Riemann surfaces. Its counterpart in Diophantine approximation follows from a recent work of Thomas J. Tucker. In this paper, we point out Tucker's result in the special case of the approximation by rational points could be obtained by doing a “translation” and applying the corresponding result with fixed target. However, we could not completely recover Tucker's result concerning the approximation by algebraic points. In the last part of this paper, cases in higher dimensions are studied. Some partial results in higher dimensions are obtained and some conjectures are raised. Received August 26, 1997; in final form June 30, 1998     

Some explicit badly approximable pairs

I consider the Diophantine approximation problem of sup-norm simultaneous rational approximation with common denominator of a pair of irrational numbers, and compute explicitly some pairs with large approximation constant. One of these pairs is the most badly approximable pair yet computed.     

An inhomogeneous Jarník theorem

We compute the generalized Hausdorff measure of sets of points in R ^s which satisfy an inhomogeneous system of Diophantine inequalities infinitely often. This provides an inhomogeneous analogue of a classical result of Jarník on simultaneous Diophantine approximation.     

Problèmes diophantiens simultanés

Simultaneous Diophantine Approximations. We study a mixed simultaneous diophantine problem, with an approximation condition and a divisibility condition. We solve this problem for quadratic numbers.     

Khinchin��s theorem for approximation by integral points on quadratic varieties

We prove an analogue of the Khinchin??s theorem for the Diophantine approximation by integer vectors lying on a quadratic variety. The proof is based on the study of a dynamical system on a homogeneous space of the orthogonal group. We show that in this system, generic trajectories of a certain geodesic flow visit a family of shrinking subsets infinitely often.     

Global Hypoellipticity and Simultaneous Approximability

A necessary and sufficient condition is given for a sum of squares operator to be globally hypoelliptic on an N-dimensional torus. This condition is expressed in terms of Diophantine approximation properties of the coefficients. The proof of the Theorem is based on L²-estimates and microlocalization.     

On The diophantine equation Fn+Fm=2a

In this paper, we find all the solutions of the title Diophantine equation in positive integer variables (n, m, a), where F_k is the kth term of the Fibonacci sequence. The proof of our main theorem uses lower bounds for linear forms in logarithms (Baker's theory) and a version of the Baker-Davenport reduction method in diophantine approximation.     

Inhomogeneous theory of dual Diophantine approximation on manifolds

The theory of inhomogeneous Diophantine approximation on manifolds is developed. In particular, the notion of nice manifolds is introduced and the divergence part of the Groshev type theory is established for all such manifolds. Our results naturally incorporate and generalize the homogeneous measure and dimension theorems for non-degenerate manifolds established to date. The results have natural applications beyond the standard inhomogeneous theory such as Diophantine approximation by algebraic integers.     

Sur la linéarité de la fonction de Artin

We give here a counter-example to an old conjecture in the theory of singularities. This conjecture is that the function that appears in the strong Artin approximation theorem is bounded by an affine function. First we study Diophantine approximation between the field of power series in several variables and its completion for the m-adic topology. We show, with an example, that there is no Liouville theorem in this case. This example gives us our counter-example (cf. théorème 1.2). As an application, we give a new proof of the fact that there is no theory of elimination of quantifiers for the field of fractions of the ring of power series in several variables.     

A simple proof of Schmidt–Summerer’s inequality

In this paper we give a simple proof of an inequality for intermediate Diophantine exponents obtained recently by W.M. Schmidt and L. Summerer.     

Gevrey hypoellipticity and solvability on the multidimensional torus of some classes of linear partial differential operators

Abstract In this paper we consider the problem of global analytic and Gevrey hypoellipticity and solvability for a class of partial differential operators on a torus. We prove that global analytic and Gevrey hypoellipticity and solvability on the torus is equivalent to certain Diophantine approximation properties. Keywords: Global hypoellipticity, Global solvability, Gevrey classes, Diophantine approximation property Mathematics Subject Classification (2000): 35D05, 46E10, 46F05, 58J99     

Invariance principles for Diophantine approximation of formal Laurent series over a finite base field

In a recent paper, the first and third author proved a central limit theorem for the number of coprime solutions of the Diophantine approximation problem for formal Laurent series in the setting of the classical theorem of Khintchine. In this note, we consider a more general setting and show that even an invariance principle holds, thereby improving upon earlier work of the second author. Our result yields two consequences: (i) the functional central limit theorem and (ii) the functional law of the iterated logarithm. The latter is a refinement of Khintchine's theorem for formal Laurent series. Despite a lot of research efforts, the corresponding results for Diophantine approximation of real numbers have not been established yet.     

Inhomogeneous Diophantine approximation on planar curves

In this paper we develop the inhomogeneous metric theory of simultaneous Diophantine approximation on planar curves. Our results naturally extend the homogeneous Khintchine and Jarník type theorems established in Beresnevich et al. (Ann Math 166(2):367–426, 2007) and Vaughan and Velani (Invent Math 166:103–124, 2006) and are the first of their kind. The key lies in obtaining essentially the best possible results regarding the distribution of ‘shifted’ rational points near planar curves.     

VALUE DISTRIBUTION THEORY AND DIOPHANTINE APPROXIMATION

In this paper, we will introduce some problems and results between Diophantine approximation and value distribution theory.     

Rational approximation of maximal commutative subgroups of $${GL(n,\mathbb{R})}$$

How to find “best rational approximations” of maximal commutative subgroups of \({GL(n,\mathbb{R})}\)? In this paper we specify this problem and make first steps in its study. It contains the classical problems of Diophantine and simultaneous approximation as particular subcases but in general is much wider. We prove estimates for n = 2 for both totally real and complex cases and give an algorithm to construct best approximations of a fixed size. In addition we introduce a relation between best approximations and sails of cones and interpret the result for totally real subgroups in geometric terms of sails.     

A note on arithmetic Diophantine series

Czechoslovak Mathematical Journal - We consider an asymptotic analysis for series related to the work of Hardy and Littlewood (1923) on Diophantine approximation, as well as Davenport. In...     

Generalization of Stanley's monster reciprocity theorem

By studying the reciprocity property of linear Diophantine systems in light of Malcev-Neumann series, we present in this paper a new approach to and a generalization of Stanley's monster reciprocity theorem. A formula for the “error term” is given in the case when the system does not have the reciprocity property. We also give an inductive proof of Stanley's reciprocity theorem for linear homogeneous Diophantine systems.     

An elementary proof of the irrationality of ζ(3)

An elementary proof of the irrationality of ζ(3) is presented. The proof is based on a two times more dense sequence of Diophantine approximations to this number than the sequence in the original proof of Apery.