114 research outputs found
On the Littlewood conjecture in fields of power series
Let \k be an arbitrary field. For any fixed badly approximable power series
in \k((X^{-1})), we give an explicit construction of continuum many
badly approximable power series for which the pair
satisfies the Littlewood conjecture. We further discuss the Littlewood
conjecture for pairs of algebraic power series
On the Littlewood conjecture in simultaneous Diophantine approximation
For any given real number with bounded partial quotients, we
construct explicitly continuum many real numbers with bounded partial
quotients for which the pair satisfies a strong form of the
Littlewood conjecture. Our proof is elementary and rests on the basic theory of
continued fractions
Palindromic continued fractions
In the present work, we investigate real numbers whose sequence of partial
quotients enjoys some combinatorial properties involving the notion of
palindrome. We provide three new transendence criteria, that apply to a broad
class of continued fraction expansions, including expansions with unbounded
partial quotients. Their proofs heavily depend on the Schmidt Subspace Theorem
On the complexity of algebraic number I. Expansions in integer bases
Let be an integer. We prove that the -adic expansion of every
irrational algebraic number cannot have low complexity. Furthermore, we
establish that irrational morphic numbers are transcendental, for a wide class
of morphisms. In particular, irrational automatic numbers are transcendental.
Our main tool is a new, combinatorial transcendence criterion
On the complexity of algebraic numbers II. Continued fractions
The continued fraction expansion of an irrational number is
eventually periodic if and only if is a quadratic irrationality.
However, very little is known regarding the size of the partial quotients of
algebraic real numbers of degree at least three. Because of some numerical
evidence and a belief that these numbers behave like most numbers in this
respect, it is often conjectured that their partial quotients form an unbounded
sequence. More modestly, we may expect that if the sequence of partial
quotients of an irrational number is, in some sense, "simple", then
is either quadratic or transcendental. The term "simple" can of course
lead to many interpretations. It may denote real numbers whose continued
fraction expansion has some regularity, or can be produced by a simple
algorithm (by a simple Turing machine, for example), or arises from a simple
dynamical system... The aim of this paper is to present in a unified way
several new results on these different approaches of the notion of
simplicity/complexity for the continued fraction expansion of algebraic real
numbers of degree at least three
On the Maillet--Baker continued fractions
We use the Schmidt Subspace Theorem to establish the transcendence of a class
of quasi-periodic continued fractions. This improves earlier works of Maillet
and of A. Baker. We also improve an old result of Davenport and Roth on the
rate of increase of the denominators of the convergents to any real algebraic
number
A problem around Mahler functions
Let be a field of characteristic zero and and be two
multiplicatively independent positive integers. We prove the following result
that was conjectured by Loxton and van der Poorten during the Eighties: a power
series satisfies both a - and a -Mahler type functional
equation if and only if it is a rational function.Comment: 52 page
M\'ethode de Mahler: relations lin\'eaires, transcendance et applications aux nombres automatiques
This paper is concerned with Mahler's method. We study in detail the
structure of linear relations between values of Mahler functions at algebraic
points. In particular, given a field , a Mahler function , and an algebraic number , , that is
not a pole for , we show that one can always determined whether the number
is transcendental or not. In the latter case, we obtain that
belong to the number fields . We also consider
some consequences of such results to a classical number theoretical problem:
the study of sequences of digits of algebraic numbers in an integer (or, more
generally, algebraic) base. Our results are based on a theorem of Philippon
[31] that we refine. We also simplify his proof.Comment: 46 pp, in Frenc
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