Field Of Formal Laurent Series Research Articles

Metric inhomogeneous Diophantine approximation on the field of formal Laurent series

Metric inhomogeneous Diophantine approximation on the field of formal Laurent series

On the relative growth rate of degrees of partial quotients in continued fractions over the field of formal Laurent series

In this paper, we study the relative growth rate of degrees of partial quotients with respect to convergents in continued fractions over the field of Laurent series. For any r > 0, the Hausdorff dimensions of the sets {x : deg An+1(x) ⩾ r · deg Qn(x) for infinitely many n} and {x : deg An+1(x) ⩾ r · deg Qn(x) for all n ⩾ 1} are determined, where An(x), n ⩾ 1, denote the partial quotients and Qn(x), n ⩾ 1, denote the denominators of the convergents.

Weyl sums in [formula omitted] with digital restrictions

Let F q be a finite field and consider the polynomial ring F q [ X ] . Let Q ∈ F q [ X ] . A function f : F q [ X ] → G , where G is a group, is called strongly Q-additive, if f ( A Q + B ) = f ( A ) + f ( B ) holds for all polynomials A , B ∈ F q [ X ] with deg B < deg Q . We estimate Weyl sums in F q [ X ] restricted by Q-additive functions. In particular, for a certain character E we study sums of the form ∑ P E ( h ( P ) ) , where h ∈ F q ( ( X −1 ) ) [ Y ] is a polynomial with coefficients contained in the field of formal Laurent series over F q and the range of P is restricted by conditions on f i ( P ) , where f i ( 1 ⩽ i ⩽ r ) are Q i -additive functions. Adopting an idea of Gel'fond such sums can be rewritten as sums of the form ∑ deg P < n E ( h ( P ) + ∑ i = 1 r R i M i f i ( A ) ) , with R i , M i ∈ F q [ X ] . Sums of this shape are treated by applying the kth iterate of the Weyl–van der Corput inequality and studying higher correlations of the functions f i . With these Weyl sum estimates we show uniform distribution results.

Metric properties and exceptional sets of β-expansions over formal Laurent series

This paper is concerned with the metric properties of β-expansions over the field of formal Laurent series. We will see that there are essential differences between β-expansions of the formal Laurent series case and the classical real case. Also the Hausdorff dimensions of some exceptional sets, with respect to the Haar measure, are determined.

Beta-expansion and continued fraction expansion over formal Laurent series

Let x ∈ I be an irrational element and n ⩾ 1 , where I is the unit disc in the field of formal Laurent series F ( ( X −1 ) ) , we denote by k n ( x ) the number of exact partial quotients in continued fraction expansion of x, given by the first n digits in the β-expansion of x, both expansions are based on F ( ( X −1 ) ) . We obtain that lim inf n → + ∞ k n ( x ) n = deg β 2 Q * ( x ) , lim sup n → + ∞ k n ( x ) n = deg β 2 Q * ( x ) , where Q * ( x ) , Q * ( x ) are the upper and lower constants of x, respectively. Also, a central limit theorem and an iterated logarithm law for { k n ( x ) } n ⩾ 1 are established.

Dynamique sur le rayon modulaire et fractions continues en caractéristique p

Let K̂ be the field of formal Laurent series in X−1 over the finite field k, and let A be the ring of polynomials in X over k. One of the main results of the paper is to give a natural coding of the (discrete) geodesic flow on the quotient of the Bruhat–Tits tree 𝕋 of PGL2(K̂) by PGL2(A), using the continued fraction expansion of the endpoints of the geodesic lines in 𝕋 (the space of ends of 𝕋 identifies with ℙ 1(K̂). In particular, the invariance of the Haar measure by the Artin transformation can be deduced from the invariance of the Bowen–Margulis measure by the geodesic flow.

Cantor sets determined by partial quotients of continued fractions of Laurent series

In this paper, two types of general sets determined by partial quotients of continued fractions over the field of formal Laurent series with coefficients from a given finite field are studied. The Hausdorff dimensions of { x : deg A n ( x ) ⩾ ϕ ( n ) , for infinitely many n } and { x : deg A n ( x ) ⩾ ϕ ( n ) , ∀ n ⩾ 1 } are determined completely, where A n ( x ) denotes the partial quotients in the continued fraction expansion (in case of Laurent series) of x and ϕ ( n ) is a positive valued function defined on natural numbers N.

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 quantitative Khintchine-Groshev type theorem over a field of formal series

An asymptotic formula which holds almost everywhere is obtained for the number of solutions to theDiophantine inequalities ‖qA − p‖ < Ψ(‖g‖), where A is an n x m matrix ( m > 1) over the field of formal Laurent series with coefficients from a finite field, and p and q are vectors of polynomials over the same finite field.

On the sum of degrees of digits occurring in continued fraction expansions of Laurent series

Bescovitch considered the Hausdorff dimensions of sets related to the sum of digits of real numbers represented in the dyadic expansions. In this paper, we provide an analogy for continued fraction expansions over the field of formal Laurent series. We also calculate the Hausdorff dimensions of sets of Laurent series which have given polynomial or exponential approximation orders. Such approximations converge faster than those of almost all Laurent series (with respect to the Haar measure).

Matsuki correspondence for the affine Grassmannian

Let $\mathcal{K}$=ℂ((t)) be the field of formal Laurent series, and let $\mathcal{O}$=ℂ[[t]] be the ring of formal power series. In this paper we present a version of the Matsuki correspondence for the affine Grassmannian Gr=G($\mathcal{K}$)/G($\mathcal{O}$) of a connected reductive complex algebraic group G. Our main statement is an anti-isomorphism between the orbit posets of two subgroups of G($\mathcal{K}$) acting on Gr. The first subgroup is the polynomial loop group LGℝ of a real form Gℝ of G; the second is the loop group K($\mathcal{K}$) of the complexification K of a maximal compact subgroup Kc of Gℝ. The orbit poset itself turns out to be simple to describe.

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.

The lagrange inversion formula on non--Archimedean fields, non--analytical form of differential and finite difference equations

The classical Lagrange inversion formula is extended to analytic andnon--analytic inversion problemson non--Archimedean fields. We give some applications to the fieldof formal Laurent series in $n$ variables, where the non--analyticinversion formula gives explicit formal solutions of generalsemilinear differential and $q$--difference equations.We will be interested in linearization problems for germs ofdiffeomorphisms (Siegel center problem) and vector fields. In addition toanalytic results, we give sufficient condition for the linearization tobelong to some Classes of ultradifferentiable germs, closed undercomposition and derivation, including Gevrey Classes. We prove thatBruno's condition is sufficient for the linearization to belong to thesame Class of the germ, whereas new conditions weaker than Bruno's one areintroduced if one allows the linearization to be less regular than thegerm. This generalizes to dimension $n> 1$ some resultsof [6]. Our formulation of the Lagrange inversion formulaby mean of trees, allows us to point out the strong similarities existingbetween the two linearization problems, formulated (essentially) with thesame functional equation. For analytic vector fields of $\mathbb C^2$ we prove aquantitative estimate of a previous qualitative resultof [25] and we compare it with a resultof [26].

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).

Automata, algebraicity and distribution of sequences of powers

Let K be a finite field of characteristic p. Let K((x)) be the field of formal Laurent series f(x) in x with coefficients in K. That is,

Engel Expansions and the Rogers–Ramanujan Identities

We study the previously developed extension of the Engel expansion to the field of Formal Laurent series. We examine three separate aspects. First we consider the algorithm in relation to the work of Ramanujan. Second we show how the algorithm can be used to prove expansions such as those found by Euler, Rogers, and Ramanujan. Finally we remark briefly on its use in acceleration of convergence.

Fq-Linear Calculus over Function Fields

We define analogues of higher derivatives forFq-linear functions over the field of formal Laurent series with coefficients inFq. This results in a formula for Taylor coefficients of aFq-linear holomorphic function, a definition of classes ofFq-linear smooth functions which are characterized in terms of coefficients of their Fourier–Carlitz expansions. A Volkenborn-type integration theory forFq-linear functions is developed; in particular, an integral representation of the Carlitz logarithm is obtained.

Approximation durch algebraische Zahlen beschr�nkten Grades im K�rper der formalen laurentreihen

Approximation by Algebraic Numbers of Bounded Degree in the Field of Formal Laurent Series. In this paper results ofWirsing andDavenport-Schmidt concerning the approximation of certain numbers by algebraic numbers of bounded degree are transferred to the field of formal power series connected with a certain non-archimedian valuation.

On the distribution of 𝑘-dimensional vectors for simple and combined Tausworthe sequences

The lattice structure of conventional linear congruential random number generators (LCGs), over integers, is well known. In this paper, we study LCGs in the field of formal Laurent series, with coefficients in the Galois field F 2 {\mathbb {F}_2} . The state of the generator (a Laurent series) evolves according to a linear recursion and can be mapped to a number between 0 and 1, producing what we call a LS2 sequence. In particular, the sequences produced by simple or combined Tausworthe generators are special cases of LS2 sequences. By analyzing the lattice structure of the LCG, we obtain a precise description of how all the k-dimensional vectors formed by successive values in the LS2 sequence are distributed in the unit hypercube. More specifically, for any partition of the k-dimensional hypercube into 2 k l {2^{kl}} identical subcubes, we can quickly compute a table giving the exact number of subcubes that contain exactly n points, for each integer n. We give numerical examples and discuss the practical implications of our results.

The unitary dual for the multiplicative group of arbitrary division algebras over local fields

Let F be a locally compact, nondiscrete, non-Archimedean field (i.e., a finite extension of Qp or a field of formal Laurent series over a finite field), and let D 2 be a division algebra of degree n over F, so that [D: F] = n . In this paper, we give a construction of all irreducible unitary representations of Dx . There does not seem to be an easy way to parametrize these representations; Theorem 5.5 is a start in that direction. The problem of determining (Dx) ^ is interesting not only in itself, but also for the light it sheds on the problem of determining all irreducible supercuspidal representations of GL, (F) . The problems are connected in two ways. The construction of supercuspidals for GL,(F) has used procedures quite similar to those used to construct (DX) ^. In previous papers, (DX) has been determined when p2 = n or p2 t n. (See [11, 10, 3, 4, 5].) The supercuspidal representations of GL,(F) are known when p t n and when n is the product of at most two distinct primes (see [12, 17, 2, 6]), and there is usually a close similarity in methods of construction to the corresponding case for division algebras. Furthermore, one important method of proving that the construction of supercuspidals is complete uses the Matching Theorem of Deligne and Kazhdan (see [8]), which sets up a natural correspondence between (Dx) ^ and the set of discrete series representations of GL, (F) . (See, e.g., [2] or [ 17] for this kind of proof.) It seems likely that the construction in this paper will also apply to give a construction of all supercuspidal representations of GL, (F). This matter will be dealt with in a future paper. (In this connection, it is worth noting the results in [15] about supercuspidals in GL, (F), n the product of two primes.) Describing the general method of the construction requires some notation. Let k be the residue class field of F, and suppose that k has q elements. Recall (from, e.g., Chapter I of [18]) that D contains an unramified extension