18 pages
English

On the values of a class of analytic functions at algebraic points

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Niveau: Supérieur, Doctorat, Bac+8
On the values of a class of analytic functions at algebraic points Boris ADAMCZEWSKI, Yann BUGEAUD and Florian LUCA Abstract. We introduce a class of analytic functions of number theo- retic interest, namely stammering functions. It has been recently proved that these functions take transcendental values at some algebraic points. In the present paper, we establish a general transcendence criterion that extends these results. Another aim is to underline the main difficulties arising from the use of the Schmidt Subspace Theorem in this context. 1. Introduction In 1844, Liouville established that the real number ∑ k≥1 10?k! is transcendental. With a similar method, it is nowadays an easy exercise to extend his result as follows (see [8], page 2). Theorem L. Let ? be an algebraic number with 0 < |?| < 1. Then, the complex number∑ k≥1 ?k! is transcendental. In other words, the analytic function f : z 7? ∑ k≥1 zk! takes transcendental values at every nonzero algebraic point in its open disc of convergence. A similar result was obtained by Mahler [6] for a much wider class of functions f , including some classical series such as F(z) = ∑ k≥0 z2 k or the Thue–Morse function T (z) = ∑ k≥0 tkzk, where tk = 1 (resp.

  • salem numbers

  • schmidt subspace

  • complex algebraic

  • mahler's method

  • every nonnegative integer

  • sequences generated

  • sequences

  • integer such


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Nombre de lectures 19
Langue English

On the values of a class of analytic functions at algebraic points

Boris A, Yann Band Florian L
DAMCZEWSKI UGEAUDUCA

Abstract.We introduce a class of analytic functions of number
theoretic interest, namely stammering functions.It has been recently proved
that these functions take transcendental values at some algebraic points.
In the present paper, we establish a general transcendence criterion that
extends these results.Another aim is to underline the main difficulties
arising from the use of the Schmidt Subspace Theorem in this context.

1. Introduction

P
−k!
In 1844, Liouville established that the real number10 istranscendental.
k≥1
With a similar method, it is nowadays an easy exercise to extend his result as follows (see
[8], page 2).
Theorem L.Letαbe an algebraic number with0<|α|<1the complex number. Then,
P
k!
αis transcendental.
k≥1
P
k!
In other words, the analytic functionf:z7→ztakes transcendental values
k≥1
at every nonzero algebraic point in its open disc of convergence.
A similar result was obtained by Mahler [6] for a much wider class of functionsf,
Pk
2
including some classical series such asF(z) =zor the Thue–Morse function
k≥0
P
k
T(z) =tkz, wheretk= 1 (resp.tk=−1), if the sum of digits of the binary
k≥0
expansion ofkis even (resp.method was subsequently refined and generalizedodd). His
by several authors including Kubota, Loxton, van der Poorten, Masser and Nishioka,
and the reader is referred to Nishioka’s Lecture Notes [8] for references.Since it always
relies on certain functional equations satisfied by the relevant series (for instance, we have
2 2
F(z) =F(z)−zandT(z) = (1−z)T(z)), Mahler’s method does not allow much
flexibility.
The heart of the proof of Theorem L and of Mahler’s results is a lower estimate for the
distance between two distinct algebraic numbers.Thanks to works by Thue, Siegel, Roth
and Schmidt, the seminal result of Liouville has been considerably improved.Thus, it is
not surprising that the use of the Schmidt Subspace Theorem in the present context yields
a considerable improvement of Theorem L. This was recently worked out for lacunary
functions by Corvaja and Zannier [5], who developed a new approach.We quote below a

2000Mathematics Subject Classification :

1

11J81, 11J61.