### BibTeX

@MISC{Waldschmidt09wordsand,

author = {Michel Waldschmidt},

title = {WORDS AND TRANSCENDENCE},

year = {2009}

}

### OpenURL

### Abstract

Abstract. Is it possible to distinguish algebraic from transcendental real numbers by considering the b-ary expansion in some base b � 2? In 1950, É. Borel suggested that the answer is no and that for any real irrational algebraic number x and for any base g � 2, the g-ary expansion of x should satisfy some of the laws that are shared by almost all numbers. For instance, the frequency where a given finite sequence of digits occurs should depend only on the base and on the length of the sequence. We are very far from such a goal: there is no explicitly known example of a triple (g, a, x), where g � 3 is an integer, a a digit in {0,...,g − 1} and x a real irrational algebraic number, for which one can claim that the digit a occurs infinitely often in the g-ary expansion of x. Hence there is a huge gap between the established theory and the expected state of the art. However, some progress has been made recently, thanks mainly to clever use of Schmidt’s subspace theorem. We review some of these results. 1. Normal Numbers and Expansion of Fundamental Constants 1.1. Borel and Normal Numbers. In two papers, the first [28] published in 1909 and the second [29] in 1950, Borel studied the g-ary expansion of real numbers, where g � 2 is a positive integer. In his second paper, he suggested that this expansion for a real irrational algebraic number should satisfy some of the laws shared by almost all numbers, in the sense of Lebesgue measure. Let g � 2 be an integer. Any real number x has a unique expansion x = a−kg k + ···+ a−1g + a0 + a1g −1 + a2g −2 + ·· ·, where k � 0 is an integer and the ai for i � −k, namely the digits of x in the expansion in base g of x, belong to the set {0, 1,...,g − 1}. Uniqueness is subject to the condition that the sequence (ai)i�−k is not ultimately constant and equal to g − 1. We write this expansion x = a−k ···a−1a0.a1a2 ·· ·. in base 10 (decimal expansion), whereas

### Citations

651 |
Algebraic Combinatorics on Words
- Lothaire
- 2002
(Show Context)
Citation Context ...llows from Mahler’s method [20] as well as from the theorem of Thue–Siegel–Roth–Ridout [57, 58, 1]. 2. Words and Automata 2.1. Words. We recall some basic facts from language theory; see for instance =-=[15, 46]-=-. We consider an alphabet A with g letters. The free monoid A∗ on A is the set of finite words a1 ...an where n � 0 and ai ∈ A for 1 � i � n. The law on A∗ is called concatenation. The number of lette... |

116 |
Automatic sequences. Theory, applications, generalizations
- Allouche, Shallit
- 2003
(Show Context)
Citation Context ...llows from Mahler’s method [20] as well as from the theorem of Thue–Siegel–Roth–Ridout [57, 58, 1]. 2. Words and Automata 2.1. Words. We recall some basic facts from language theory; see for instance =-=[15, 46]-=-. We consider an alphabet A with g letters. The free monoid A∗ on A is the set of finite words a1 ...an where n � 0 and ai ∈ A for 1 � i � n. The law on A∗ is called concatenation. The number of lette... |

103 | On the Rapid Computation of Various Polylogarithmic Constants
- Bailey, Borwein, et al.
(Show Context)
Citation Context ...t points or is uniformly distributed modulo 1. A connection with special values of G-functions has been noted by Lagarias [44]. In his paper, Lagarias defines BBP numbers, with reference to the paper =-=[21]-=- by Bailey, Borwein and Plouffe, as numbers of the form ∑ n�1 j=1 p(n) q(n) g−n ,452 MICHEL WALDSCHMIDT hal-00407221, version 1 - 24 Jul 2009 where g � 2 is an integer, p and q relatively prime polyn... |

93 |
Lesprobabilites denombrables et leurs applications arithmetiques
- Borel
- 1909
(Show Context)
Citation Context ...s mainly to clever use of Schmidt’s subspace theorem. We review some of these results. 1. Normal Numbers and Expansion of Fundamental Constants 1.1. Borel and Normal Numbers. In two papers, the first =-=[28]-=- published in 1909 and the second [29] in 1950, Borel studied the g-ary expansion of real numbers, where g � 2 is a positive integer. In his second paper, he suggested that this expansion for a real i... |

93 |
Applied Combinatorics on Words. Encyclopedia of Mathematics
- LOTHAIRE
- 2005
(Show Context)
Citation Context ...ollows from Mahler’s method [20] as well as from the theorem of Thue-Siegel-RothRidout [57, 58, 1]. 2. Words and Automata 2.1. Words. We recall some basic facts from language theory; see for instance =-=[15, 46]-=-. We consider an alphabet A with g letters. The free monoid A ∗ on A is the set of finite words a1 . . .an where n � 0 and ai ∈ A for 1 � i � n. The law on A ∗ is called concatenation. The number of l... |

92 |
Rational approximations to algebraic numbers
- Roth
- 1955
(Show Context)
Citation Context ...ilable, which rest either on Mahler’s method, to be discussed in Section 3.1, or on the approximation theorem of Thue–Siegel–Roth–Ridout, to be discussed in Section 4; see also [61, Section 1.6], and =-=[57, 58, 1]-=-. Another consequence of Theorem 1.1 is the transcendence of the number ∑ g −Fn , (4) n�0 for any integer g � 2, where (Fn)n�0 is the Fibonacci sequence, with F0 =0,F1 =1 and Fn+1 = Fn + Fn−1 for n � ... |

85 |
Uniform tag sequences
- Cobham
- 1972
(Show Context)
Citation Context ...=a, 101[i] =b, 110[i] =b, .... Next define f(i) =0,f(a) =1andf(b) = 0. The output sequence a0a1a2 ...= 01101000100000001000 ... is given by { 1 if n isapowerof2, an = 0 otherwise. According to Cobham =-=[39]-=-, automatic sequences have complexity p(m) =O(m); see also [15, Section 10.3]. Also, automatic sequences are the same as uniform morphic sequences; see for instance [15, Section 6.3] and [8, Theorem 4... |

56 |
The construction of decimals normal in the scale of ten
- Champernowne
- 1933
(Show Context)
Citation Context ...ul 2009 An example of a 2-normal number is the binary Champernowne number, obtained by concatenation of the sequence of integers 0. 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 ...; see =-=[35, 54, 22]-=-. A closed formula for this number is ∑ k2 −ck k∑ , ck = k + ⌊log2 j⌋; k�1 see [22, p. 183]. Another example is given by Korobov, Stoneham, and others: if a and g are coprime integers greater than 1, ... |

55 |
Approximation by algebraic numbers, Cambridge Tracts
- Bugeaud
- 2004
(Show Context)
Citation Context ...geaud [6] have obtained transcendence measures for automatic numbers. They show that automatic irrational numbers are either S- orT - numbers in Mahler’s classification of transcendental numbers; see =-=[30]-=-. This is a partial answer to a conjecture of Becker [8] which states that all automatic irrational numbers are S-numbers. 5. Continued Fractions We discussed above some Diophantine problems which are... |

54 |
Information and Randomness: An Algorithmic Perspective, 2nd Edition, Revised and Extended
- Calude
- 2002
(Show Context)
Citation Context ..., obtained by concatenation of the sequence of prime numbers. As pointed out to the author by Tanguy Rivoal, a definition of what it means for a number to have a random sequence of digits is given in =-=[33]-=-, the first concrete example being Chaitin’s omega number which is the halting probability of a universal selfdelimiting computer with null-free data. This number, which is normal and transcendental, ... |

51 | On the random character of fundamental constant expansions
- Bailey, Crandall
(Show Context)
Citation Context ...ul 2009 An example of a 2-normal number is the binary Champernowne number, obtained by concatenation of the sequence of integers 0. 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 ...; see =-=[35, 54, 22]-=-. A closed formula for this number is ∑ k2 −ck k∑ , ck = k + ⌊log2 j⌋; k�1 see [22, p. 183]. Another example is given by Korobov, Stoneham, and others: if a and g are coprime integers greater than 1, ... |

47 |
Suites algébriques, automates et substitutions
- Christol, Kamae, et al.
- 1980
(Show Context)
Citation Context ...ng to Cobham, a sequence can be simultaneously k-automatic and ℓ-automatic with k and ℓ multiplicatively independent if and only if it is ultimately periodic. Christol, Kamae, Mendès France and Rauzy =-=[37]-=- have shown that a power series with coefficients in a finite set can be algebraic over two different finite fields if and only if it is rational. In this context Theorem 2.3 implies the following sta... |

45 |
Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen
- Mahler
- 1929
(Show Context)
Citation Context ... algebraic number contains infinitely many occurrences of overlaps, where an overlap is a pattern of the form xXxXx, where x is a letter and X a word. 3. Analytic Methods 3.1. Mahler’s Method. Mahler =-=[48]-=- initiated a new transcendence method in 1929 for studying values of functions satisfying certain functional equations. His work was not widely known and in 1969 he published [49] which was the source... |

40 | Transcendence of Sturmian or morphic continued fractions
- Allouche, Davison, et al.
(Show Context)
Citation Context ... the Prouhet–Thue–Morse continued fraction; see also [15, Section 13.7]. There are further papers by Liardet and Stambul in 2000 and by Baxa in 2004. In 2001, Allouche, Davison, Queffélec and Zamboni =-=[13]-=- proved the transcendence of Sturmian continued fractions, namely continued fractions whose sequence of partial quotients is Sturmian. The transcendence of the Rudin–Shapiro and of the Baum–Sweet cont... |

38 |
Automatic Sequences: Theory
- Allouche, Shallit
- 2003
(Show Context)
Citation Context ...ollows from Mahler’s method [20] as well as from the theorem of Thue-Siegel-RothRidout [57, 58, 1]. 2. Words and Automata 2.1. Words. We recall some basic facts from language theory; see for instance =-=[15, 46]-=-. We consider an alphabet A with g letters. The free monoid A ∗ on A is the set of finite words a1 . . .an where n � 0 and ai ∈ A for 1 � i � n. The law on A ∗ is called concatenation. The number of l... |

37 |
Ensembles presques périodiques k-reconnaissables, Theor
- Christol
- 1979
(Show Context)
Citation Context ... Theorem 2.3 is a new, combinatorial transcendence criterion obtained by Adamczewski, Bugeaud and Luca [7] as an application of Schmidt’s subspace theorem; see Theorem 4.4 and [60]. In 1979, Christol =-=[36]-=- proved that when p is a prime and f(X) = ∑ ukX k ∈ Fp[[X]] k�1 a power series with coefficients in the finite field Fp, then f is algebraic if and only if the sequence of coefficients (uk)k�1 can be ... |

37 |
Trancedence of numbers with a low complexity expansions
- Ferenczi, Mauduit
- 1997
(Show Context)
Citation Context ...hue–Morse sequence, see [18]. 2.4. Complexity of the g-ary Expansion of an Algebraic Number. The transcendence of a number whose sequence of digits is Sturmian has been proved by Ferenczi and Mauduit =-=[42]-=- in 1997. The point is that such sequences contain sequences of digits which bear similarities, and yields the existence of very sharp rational approximations which do not exist for algebraic numbers.... |

33 | On the complexity of algebraic numbers I. Expansions in integer bases
- Adamczewski, Bugeaud
(Show Context)
Citation Context ...eal algebraic numbers, in particular by Allouche and Zamboni in 1998, Risley and Zamboni in 2000, and Adamczewski and Cassaigne in 2003. For a survey, see [3]. The main recent result is the following =-=[5]-=-. 0460 MICHEL WALDSCHMIDT hal-00407221, version 1 - 24 Jul 2009 Theorem 2.3 (Adamczewski and Bugeaud, 2007). The complexity p(m) of the g-ary expansion of a real irrational algebraic number satisfies... |

30 |
Sur la complexité des nombres algébriques
- Adamczewski, Bugeaud, et al.
(Show Context)
Citation Context ...n irrational real number x is automatic, then x is transcendental. The main tool for the proof of Theorem 2.3 is a new, combinatorial transcendence criterion obtained by Adamczewski, Bugeaud and Luca =-=[7]-=- as an application of Schmidt’s subspace theorem; see Theorem 4.4 and [60]. In 1979, Christol [36] proved that when p is a prime and f(X) = ∑ ukX k ∈ Fp[[X]] k�1 a power series with coefficients in th... |

28 |
Mahler functions and transcendence
- Nishioka
- 1996
(Show Context)
Citation Context ...hod in 1929 for studying values of functions satisfying certain functional equations. His work was not widely known and in 1969 he published [49] which was the source of a revival of this method; see =-=[47, 52]-=-. A first example [52, Theorem 1.1.2] is the function f(z) = ∑ z −dn , d � 2, n�0 which satisfies the functional equationf(z d )+z = f(z) for |z| < 1. Mahler proved that it takes transcendental values... |

26 | Eds.), Introduction to Algebraic Independence Theory - Nesterenko, Philippon |

26 | Einführung in die Transzendenten Zahlen - Schneider - 1957 |

25 |
An example of a computable absolutely normal number
- Becher, Figueira
- 2002
(Show Context)
Citation Context ...As shown by Borel [28], almost all numbers, in the sense of Lebesgue measure, are normal. Examples of computable normal numbers have been constructed by Sierpinski, Lebesgue, Becher and Figueira; see =-=[24]-=-. However, the known algorithms to compute such examples are fairly complicated; indeed, “ridiculously exponential”, according to [24].WORDS AND TRANSCENDENCE 451 hal-00407221, version 1 - 24 Jul 200... |

24 | Experimental Mathematics: Recent Developments And Future Outlook
- Bailey, Borwein
- 2001
(Show Context)
Citation Context ... q(n) g−n ,452 MICHEL WALDSCHMIDT hal-00407221, version 1 - 24 Jul 2009 where g � 2 is an integer, p and q relatively prime polynomials in Z[X] with q(n) ̸= 0 for n � 1. Here are a few examples from =-=[19]-=-. Since ∑ n�1 1 n xn = − log(1 − x) and ∑ n�1 n�1 1 2n − 1 x2n−1 = 1 1+x log 2 1 − x , it follows that log 2 is a BBP number in base 2 as well as in base 9, since log 2 = ∑ 1 n 2−n = ∑ 6 2n − 1 3−2n .... |

24 |
les chiffres décimaux de √ 2 et divers problèmes de probabilités en chaîne
- Borel, Sur
- 1950
(Show Context)
Citation Context ...bspace theorem. We review some of these results. 1. Normal Numbers and Expansion of Fundamental Constants 1.1. Borel and Normal Numbers. In two papers, the first [28] published in 1909 and the second =-=[29]-=- in 1950, Borel studied the g-ary expansion of real numbers, where g � 2 is a positive integer. In his second paper he suggested that this expansion for a real irrational algebraic number should satis... |

23 | Complexity of sequences defined by billiards in the cube
- Arnoux, Mauduit, et al.
- 1994
(Show Context)
Citation Context ...edlund, a word of minimal complexity p(m) =m + 1 is called a Sturmian word. Sturmian words are those which encode with two letters the orbits of square billiard starting with an irrational angle; see =-=[16, 17, 62]-=-. It is easy to check that on the alphabet {a, b}, a Sturmian word w is characterized by the property that for each m � 1, there is exactly one factor v of w of length m for which both va and vb are f... |

23 |
On the binary expansions of algebraic numbers
- Bailey, Borwein, et al.
(Show Context)
Citation Context ...k∑ , ck = k + ⌊log2 j⌋; k�1 see [22, p. 183]. Another example is given by Korobov, Stoneham, and others: if a and g are coprime integers greater than 1, then ∑ n�0 a −n g −an is normal in base g; see =-=[20]-=-. A further example, due to Copeland and Erdős in 1946, of a normal number in base 10 is 0.23571113171923 ..., obtained by concatenation of the sequence of prime numbers. As pointed out to the author ... |

22 |
Continued fractions of algebraic power series in characteristic 2
- Baum, Sweet
- 1976
(Show Context)
Citation Context ...l such x, or that they are unbounded for all such x. The common expectation seems to be that they are never bounded. The situation in finite characteristic is quite different. In 1976, Baum and Sweet =-=[23]-=- constructed a formal series which is cubic over the field F2(X), the continued fractions of which have partial quotients of bounded degree. We give here only a very short historical account. The very... |

18 |
Some new applications of the Subspace Theorem
- Corvaja, Zannier
(Show Context)
Citation Context ...g |L0,ℓ(x0,x1)|ℓ = q −1 , |L1,∞(x0,x1)|∞ = |qα − p|, ∏ |L1,ℓ(x0,x1)|ℓ = |p|ℓ � 1. ℓ|g Further applications of the Subspace theorem to transcendence questions have been obtained by Corvaja and Zannier =-=[40]-=-. hal-00407221, version 1 - 24 Jul 2009 4.2. Irrationality and Transcendence Measures. The previous results can be made effective in order to reach irrationality measures or transcendence measures for... |

17 |
On the Hartmanis-Stearns problem for a class of tag machines
- Cobham
- 1968
(Show Context)
Citation Context ...rsion 1 - 24 Jul 2009 Theorem 2.3 (Adamczewski and Bugeaud, 2007). The complexity p(m) of the g-ary expansion of a real irrational algebraic number satisfies lim inf m→∞ p(m) m =+∞. In 1968, 1 Cobham =-=[38]-=- claimed that automatic irrational numbers are transcendental. This follows from Theorem 2.3, since automatic numbers have a complexity O(m). Corollary 2.4 (Conjecture of Cobham). If the sequence of d... |

15 |
Nouveaux résultats de transcendance de réels à développement non aléatoire, Gaz
- Allouche
- 2000
(Show Context)
Citation Context ...ximations which do not exist for algebraic numbers. It follows from their work that the complexity of the g-ary expansion of every irrational algebraic number satisfies lim inf(p(m) − m) =+∞; m→∞ see =-=[10]-=-. The main tool for the proof is a p-adic version of the Thue–Siegel–Roth theorem due to Ridout in 1957; see Theorem 4.3 below as well as [3]. Several papers have been devoted to the study of the comp... |

15 |
Aspects de l’indépendance algébrique en caractéristique non nulle. [d’après
- Pellarin
- 2007
(Show Context)
Citation Context ...ting development of Mahler’s method which we only allude to here without entering the subject. It is due to Denis and deals with transcendence problems in finite characteristic; see Pellarin’s report =-=[53]-=- in the Bourbaki Seminar. 3.2. Nesterenko’s Theorem and Consequences. The transcendence of the Liouville-Fredholm number (7) below was studied only ten years ago by Bertrand in 1997, and Duverney, Nis... |

15 |
Transcendance des fractions continues de Thue–Morse
- Queffélec
- 1998
(Show Context)
Citation Context ...ations are due to Baker in 1962 and 1964. The approximation results on real numbers by quadratic numbers due to Schmidt in 1967 are a main tool for the next steps by Davison in 1989, and by Queffélec =-=[55]-=- who established in 1998 the transcendence of the Prouhet–Thue–Morse continued fraction; see also [15, Section 13.7]. There are further papers by Liardet and Stambul in 2000 and by Baxa in 2004. In 20... |

14 |
Approximation simultanée d’un nombre et de son carré
- Roy
(Show Context)
Citation Context ...on expansion of an algebraic number of degree at least three cannot be generated by a binary morphism. 5.2. The Fibonacci Continued Fraction. The Fibonacci word discussed in Section 2.3.1 enabled Roy =-=[59]-=- to construct transcendental real numbers ξ for which ξ and ξ2 are surprisingly well simultaneously and uniformly approximated by rational numbers. Recall once more that Φ denotes the golden number, s... |

10 | Diophantine properties of real numbers generated by finite automata
- Adamczewski, Cassaigne
(Show Context)
Citation Context ...ity and Transcendence Measures. The previous results can be made effective in order to reach irrationality measures or transcendence measures for automatic numbers. In 2006, Adamczewski and Cassaigne =-=[8]-=- solved a conjecture of Shallit in 1999 by proving that the sequence of g-ary digits of a Liouville number cannot be generated by a finite automaton. They obtained irrationality measures for automatic... |

9 |
Remarks on a paper by W
- Mahler
- 1969
(Show Context)
Citation Context ...ler’s Method. Mahler [48] initiated a new transcendence method in 1929 for studying values of functions satisfying certain functional equations. His work was not widely known and in 1969 he published =-=[49]-=- which was the source of a revival of this method; see [47, 52]. A first example [52, Theorem 1.1.2] is the function f(z) = ∑ z −dn , d � 2, n�0 which satisfies the functional equationf(z d )+z = f(z)... |

8 |
Transcendance “à la Liouville” de certains nombres réels
- Adamczewski
(Show Context)
Citation Context ...ilable, which rest either on Mahler’s method, to be discussed in Section 3.1, or on the approximation theorem of Thue–Siegel–Roth–Ridout, to be discussed in Section 4; see also [61, Section 1.6], and =-=[57, 58, 1]-=-. Another consequence of Theorem 1.1 is the transcendence of the number ∑ g −Fn , (4) n�0 for any integer g � 2, where (Fn)n�0 is the Fibonacci sequence, with F0 =0,F1 =1 and Fn+1 = Fn + Fn−1 for n � ... |

8 | On the complexity of algebraic numbers, II. Continued fractions, Acta Math
- Adamczewski, Bugeaud
- 2008
(Show Context)
Citation Context ...artial quotients is Sturmian. The transcendence of the Rudin–Shapiro and of the Baum–Sweet continued fractions was proved in 2005 by Adamczewski, Bugeaud and Davison. In 2005, Adamczewski and Bugeaud =-=[2]-=- showed that the continued fraction expansion of an algebraic number of degree at least three cannot be generated by a binary morphism. 5.2. The Fibonacci Continued Fraction. The Fibonacci word discus... |

8 |
k-Regular power series and Mahler-type functional equations
- Becker
- 1994
(Show Context)
Citation Context ...e. After Cobham [39], Loxton and van der Poorten also tried 2 in 1982 and 1988 to use Mahler’s method to prove Cobham’s conjecture, now Corollary 2.4 of Theorem 2.3 of Adamczewski and Bugeaud. Becker =-=[25]-=- pointed out in 1994 that Mahler’s method yields only a weaker result so far, that for any given non-eventually periodic automatic sequence u =(u1,u2,u3 ...), the real number ∑ ukg −k k�1 is transcend... |

8 |
der Poorten, Arithmetic properties of certain functions in several variables III
- Loxton, van
- 1977
(Show Context)
Citation Context ...hod in 1929 for studying values of functions satisfying certain functional equations. His work was not widely known and in 1969 he published [49] which was the source of a revival of this method; see =-=[47, 52]-=-. A first example [52, Theorem 1.1.2] is the function f(z) = ∑ z −dn , d � 2, n�0 which satisfies the functional equationf(z d )+z = f(z) for |z| < 1. Mahler proved that it takes transcendental values... |

6 |
La suite de Thue-Morse et la catégorie Rec
- Bacher
(Show Context)
Citation Context ...efined recursively by an =1ifn isapowerof2,and a 2 k +a =1− a 2 k −a, 1 � a<2 k ; see [15, Example 5.1.6]. For a connection between the paper folding sequence and the Prouhet–Thue–Morse sequence, see =-=[18]-=-. 2.4. Complexity of the g-ary Expansion of an Algebraic Number. The transcendence of a number whose sequence of digits is Sturmian has been proved by Ferenczi and Mauduit [42] in 1997. The point is t... |

6 | The many faces of the Subspace Theorem [after Adamczewski
- Bilu
(Show Context)
Citation Context ... very special cases of Schmidt’s subspace theorem [60], established in 1972, together with its p-adic extension by Schlickewei in 1976. We state only a simplified version; see Bilu’s Bourbaki lecture =-=[27]-=- for references and further recent achievements based on this fundamental result. For x =(x0,...,xm−1) ∈ Zm , let |x| = max{|x0|,...,|xm−1|}. Theorem 4.4 (Schmidt’s subspace theorem). Let m � 2 be an ... |

5 |
On the b-ary expansion of an algebraic
- Bugeaud
(Show Context)
Citation Context ...h that the number of 1’s among the first N digits in the binary expansion of x is at least CN1/d .WORDS AND TRANSCENDENCE 453 Further results related to Theorem 1.1 are given by Rivoal [56], Bugeaud =-=[31]-=-, and Bugeaud and Evertse [32]. As pointed out by Bailey, Borwein, Crandall and Pomerance, it follows from Theorem 1.1 that for each d � 2, the number ∑ hal-00407221, version 1 - 24 Jul 2009 n�0 2 −dn... |

4 |
Mesures de transcendance et aspects quantitatifs de la méthode de Thue-Siegel-Roth-Schmidt, tapuscript
- Adamczewski, Bugeaud
(Show Context)
Citation Context ...ch exponents, as pointed out to the author by Adamczewski. The irrationality exponent for the automatic number associated with σ(0)=0 n 1 and σ(1)=1 n 0isat least n. Recently, Adamczewski and Bugeaud =-=[6]-=- have obtained transcendence measures for automatic numbers. They show that automatic irrational numbers are either S- orT - numbers in Mahler’s classification of transcendental numbers; see [30]. Thi... |

4 |
Certain classes of transcendental numbers
- Danilov
- 1972
(Show Context)
Citation Context ... a characterization of Sturmian words. Concerning the sequence (vn)n�0 =(0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0,...), (5) derived from the Fibonacci word on the alphabet {0, 1}, a result of Danilov =-=[41]-=- in 1972 states that for all integers g � 2, the number ∑ vng −n n�0 is transcendental; see also [11, Theorem 4.2]. Proposition 2.2. The Fibonacci word is not automatic. Proposition 2.2 follows from a... |

3 | On the normality of arithmetical constants
- Lagarias
(Show Context)
Citation Context ...gyn + p(n) mod 1. q(n) Then the sequence (yn)n�1 either has finitely many limit points or is uniformly distributed modulo 1. A connection with special values of G-functions has been noted by Lagarias =-=[44]-=-. In his paper, Lagarias defines BBP numbers, with reference to the paper [21] by Bailey, Borwein and Plouffe, as numbers of the form ∑ n�1 j=1 p(n) q(n) g−n ,452 MICHEL WALDSCHMIDT hal-00407221, ver... |

3 |
On the vector space of the automatic reals
- Lehr, Shallit, et al.
- 1996
(Show Context)
Citation Context ...e of a real number x>0 which is normal in base g and for which 1/x is not normal in base g. • Give an explicit example of a real number x>0 which is normal and for which 1/x is not normal. Remark. In =-=[45]-=-, there is a construction of an automatic number, the inverse of which is not automatic. This answers by anticipation [15, Section 13.9, Problem 2]. From the open problems in [15, Section 13.9], we se... |

3 | On the bits counting function of real numbers
- Rivoal
(Show Context)
Citation Context ...) � B(x, n)B(y, n) + log2⌊x + y +1⌋. If x is positive and irrational, then for each integer A>0, the bound ( A B(x, n)B x ,n ) ⌊ ( � n − 1 − log2 x + A x +1 )⌋ holds for all sufficiently large n; see =-=[20, 56]-=-. A consequence is that if a and b are two integers, both at least 2, then none of the powers of the transcendental number ξ = ∑ a −bn n�1 is simply normal in base 2. Also the lower bound B( √ 2,n) � ... |

2 | On the decimal expansion of algebraic numbers
- Adamczewski, Bugeaud
(Show Context)
Citation Context ...algebraic number satisfies lim inf(p(m) − m) =+∞; m→∞ see [10]. The main tool for the proof is a p-adic version of the Thue–Siegel–Roth theorem due to Ridout in 1957; see Theorem 4.3 below as well as =-=[3]-=-. Several papers have been devoted to the study of the complexity of the g-ary expansions of real algebraic numbers, in particular by Allouche and Zamboni in 1998, Risley and Zamboni in 2000, and Adam... |

2 |
and p-adic expansions involving symmetric patterns
- Real
(Show Context)
Citation Context ...that the number π is not 2-automatic; see (2).468 MICHEL WALDSCHMIDT hal-00407221, version 1 - 24 Jul 2009 We conclude this chapter with one last open problem, attributed to Mahler; see for instance =-=[4]-=-: • Let (en)n�1 be an infinite sequence over {0, 1} that is not ultimately periodic. Is it true that at least one of the two numbers ∑ en2 −n and ∑ en3 −n n�1 is transcendental? From Conjecture 1.1 wi... |