Inhomogeneous Diophantine approximation over the field of formal Laurent series

De Mathan [B. de Mathan, Approximations diophantiennes dans un corps local, Bull. Soc. Math. France, Suppl. Mém. 21 (1970)] proved that Khintchine's theorem on homogeneous Diophantine approximation has an analogue in the field of formal Laurent series. Kristensen [S. Kristensen, On the well-approximable matrices over a field of formal series, Math. Proc. Cambridge Philos. Soc. 135 (2003) 255–268] extended this metric theorem to systems of linear forms and gave the exact Hausdorff dimension of the corresponding exceptional sets. In this paper, we study the inhomogeneous Diophantine approximation over a field of formal Laurent series, the analogue Khintchine's theorem and Jarnik–Besicovitch theorem are proved.     

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.     

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

In a recent paper, Kim and Nakada proved an analogue of Kurzweilʼs 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.     

形式Laurent级数域上交错Oppenheim展开的研究

该文介绍了形式Laurent级数域上交错Oppenheim展开的算法,得到了该展开中数字的强(弱)大数定理、中心极限定理和重对数率,并且研究了这些级数部分和的逼近的度.     

An analogue of a theorem of Szüsz for formal Laurent series over finite fields

About 40 years ago, Szüsz proved an extension of the well-known Gauss-Kuzmin theorem. This result played a crucial role in several subsequent papers (for instance, papers due to Szüsz, Philipp, and the author). In this note, we provide an analogue in the field of formal Laurent series and outline applications to the metric theory of continued fractions and to the metric theory of diophantine approximation.     

Approximation orders of formal Laurent series by Oppenheim rational functions

We study formal Laurent series which are better approximated by their Oppenheim convergents. We calculate the Hausdorff dimensions of sets of Laurent series which have given polynomial or exponential approximation orders. Such approximations are faster than the approximation of typical Laurent series (with respect to the Haar measure).     

On Metric Diophantine Approximation in the Field of Formal Laurent Series

B. deMathan (1970, Bull. Soc. Math. France Supl. Mem.21) proved that Khintchine’s Theorem has an analogue in the field of formal Laurent series. First, we show that in case of only one inequality this result can also be obtained by continued fraction theory. Then, we are interested in the number of solutions and show under special assumptions that one gets a central limit theorem, a law of iterated logarithm and an asymptotic formula. This is an analogue of a result due to W. J. LeVeque (1958, Trans. Amer. Math. Soc.87, 237–260). The proof is based on probabilistic results for formal Laurent series due to H. Niederreiter (1988, in Lecture Notes in Computer Science, Vol. 330, pp. 191–209, Springer-Verlag, New York/Berlin).     

On the group extension of the transformation associated to non-archimedean continued fractions

Summary Let F_q be a finite field with q elements. We consider formal Laurent series of F_q -coefficients with their continued fraction expansions by F_q -polynomials. We prove some arithmetic properties for almost every formal Laurent series with respect to the Haar measure. We construct a group extension of the non-archimedean continued fraction transformation and show its ergodicity. Then we get some results as an application of the individual ergodic theorem. We also discuss the convergence rate for limit behaviors.     

Formal Laurent series in several variables

We explain the construction of fields of formal infinite series in several variables, generalizing the classical notion of formal Laurent series in one variable. Our discussion addresses the field operations for these series (addition, multiplication, and division), the composition, and includes an implicit function theorem.     

On the Central Limit Theorem for Non-Archimedean Diophantine Approximations

We consider a Diophantine inequality:on the set of formal Laurent series of negative degree. We show that under these two conditions: (i) qⁿ(n) is a monotone non-increasing and (ii) _nqⁿ(n)=, a central limit theorem holds for the number of solutions. The proof is based on the construction of a non-stationary one dependent process associated with the Diophantine inequality.Mathematics Subject Classification (2000): 11J61, 11K60     

On formal Laurent series

Several kinds of formal Laurent series have been introduced with some restrictions so far. This paper systematically sets up a natural definition and structure of formal Laurent series without those restrictions, including introducing a multiplication between formal Laurent series. This paper also provides some results on the algebraic structure of the space of formal Laurent series, denoted by \mathbbL\mathbb{L}. By means of the results of the generalized composition of formal power series, we define a composition of a Laurent series with a formal power series and provide a necessary and sufficient condition for the existence of such compositions. The calculus about formal Laurent series is also introduced.     

A Survey of Diophantine Approximationin Fields of Power Series

 The fields of power series (or perhaps better called formal numbers) are analogues of the field of real numbers. Many questions in number theory which have been studied in the setting of the real numbers can be transposed to the setting of the power series. The study of rational approximation to algebraic real numbers has been intensively developped starting from the middle of the nineteenth century with the work of Liouville up to the celebrated theorem of Roth established in 1955. In the last thirty years, several mathematicians have studied diophantine approximation in fields of power series. We present here a summary of the present knowledge on this subject, emphasizing the analogies and differences with the situation in the real numbers case. (Received 20 January 2000)     

A Cartan’s Second Main Theorem Approach in Nevanlinna Theory

In 2002, in the paper entitled “A subspace theorem approach to integral points on curves”, Corvaja and Zannier started the program of studying integral points on algebraic varieties by using Schmidt’s subspace theorem in Diophantine approximation. Since then, the program has led a great progress in the study of Diophantine approximation. It is known that the counterpart of Schmidt’s subspace in Nevanlinna theory is H. Cartan’s Second Main Theorem. In recent years, the method of Corvaja and Zannier has been adapted by a number of authors and a big progress has been made in extending the Second Main Theorem to holomorphic mappings from C into arbitrary projective variety intersecting general divisors by using H. Cartan’s original theorem. We call such method “a Cartan’s Second Main Theorem approach”. In this survey paper, we give a systematic study of such approach, as well as survey some recent important results in this direction including the recent work of the author with Paul Voja.     

A Survey of Diophantine Approximationin Fields of Power Series

 The fields of power series (or perhaps better called formal numbers) are analogues of the field of real numbers. Many questions in number theory which have been studied in the setting of the real numbers can be transposed to the setting of the power series. The study of rational approximation to algebraic real numbers has been intensively developped starting from the middle of the nineteenth century with the work of Liouville up to the celebrated theorem of Roth established in 1955. In the last thirty years, several mathematicians have studied diophantine approximation in fields of power series. We present here a summary of the present knowledge on this subject, emphasizing the analogies and differences with the situation in the real numbers case.     

Central limit theorems and invariance principles for Lorenz attractors

We prove statistical limit laws for Hölder observationsof the Lorenz attractor, and more generally for geometric Lorenzattractors. In particular, we prove the almost sure invarianceprinciple (approximation by Brownian motion). Standard consequencesof this result include the central limit theorem, the law ofthe iterated logarithm, and the functional versions of theseresults.     

Inverse polynomial expansions of Laurent series

An algorithm is introduced, and shown to lead to various unique series expansions of formal Laurent series, as the sums of reciprocals of polynomials. The degrees of approximation by the rational functions which are the partial sums of these series are investigated. The types of series corresponding to rational functions themselves are also characterized.     

On absolute Galois splitting fields of central simple algebras

A splitting field of a central simple algebra is said to be absolute Galois if it is Galois over some fixed subfield of the centre of the algebra. The paper proves an existence theorem for such fields over global fields with enough roots of unity. As an application, all twisted function fields and all twisted Laurent series rings over symbol algebras (or p-algebras) over global fields are crossed products. An analogous statement holds for division algebras over Henselian valued fields with global residue field.The existence of absolute Galois splitting fields in central simple algebras over global fields is equivalent to a suitable generalization of the weak Grunwald-Wang theorem, which is proved to hold if enough roots of unity are present. In general, it does not hold and counter examples have been used in noncrossed product constructions. This paper shows in particular that a certain computational difficulty involved in the construction of explicit examples of noncrossed product twisted Laurent series rings cannot be avoided by starting the construction with a symbol algebra.     

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 functional central limit theorem for diffusions on periodic submanifolds of ℝ N

We prove a functional central limit theorem for diffusions on periodic sub- manifolds of ℝ^N. The proof is an adaptation of a method presented in [BenLioPap] and [Bha] for proving functional central limit theorems for diffusions with periodic drift vectorfields. We then apply the central limit theorem in order to obtain a recurrence and a transience criterion for periodic diffusions. Other fields of applications could be heat-kernel estimates, similar to the ones obtained in [Lot].Mathematics Subject Classification (2000): 35B27, 60F05, 58J65The author wants to express his gratitude toward the National Cheng Kung University in Tainan (Taiwan) for its kind hospitality.     

Remarks on Hilbert series of graded modules over polynomial rings

In this article we discuss a result on formal Laurent series and some of its implications for Hilbert series of finitely generated graded modules over standard-graded polynomial rings: For any integer Laurent function of polynomial type with non-negative values the associated formal Laurent series can be written as a sum of rational functions of the form ${\frac{Q_j(t)}{(1-t)^j}}$ , where the numerators are Laurent polynomials with non–negative integer coefficients. Hence any such series is the Hilbert series of some finitely generated graded module over a suitable polynomial ring ${\mathbb{F}[X_1 , \ldots , X_n]}$ . We give two further applications, namely an investigation of the maximal depth of a module with a given Hilbert series and a characterization of Laurent polynomials which may occur as numerator in the presentation of a Hilbert series as a rational function with a power of (1 ? t) as denominator.