Nombres réels de complexité sous-linéaire : mesures d'irrationalité et de transcendance

Adamczewski, Boris; Bugeaud, Yann

doi:10.1515/crelle.2011.061

Cited by 22 publications

(42 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As observed in [1] (see also Section 8.5 of [17]), it follows from the results of [14] and [4] that, for any integer b ≥ 2 and for any quasi-Sturmian word s over {0, 1, . .…”

Section: Rational Approximation Of Quasi-sturmian Numbers and Applicamentioning

confidence: 52%

A new complexity function, repetitions in Sturmian words, and irrationality exponents of Sturmian numbers

Bugeaud¹,

Kim

2018

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. We introduce and study a new complexity function in combinatorics on words, which takes into account the smallest second occurrence time of a factor of an infinite word. We characterize the eventually periodic words and the Sturmian words by means of this function. Then, we establish a new result on repetitions in Sturmian words and show that it is best possible. Let b ≥ 2 be an integer. We deduce a lower bound for the irrationality exponent of real numbers whose sequence of b-ary digits is a Sturmian sequence over {0, 1, . . . , b − 1} and we prove that this lower bound is best possible. As an application, we derive some information on the b-ary expansion of log(1 + 1 a ), for any integer a ≥ 34.

show abstract

“…As observed in [1] (see also Section 8.5 of [17]), it follows from the results of [14] and [4] that, for any integer b ≥ 2 and for any quasi-Sturmian word s over {0, 1, . .…”

Section: Rational Approximation Of Quasi-sturmian Numbers and Applicamentioning

confidence: 52%

A new complexity function, repetitions in Sturmian words, and irrationality exponents of Sturmian numbers

Bugeaud¹,

Kim

2018

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Recent developments have shown that the use of quantitative versions of the Schmidt Subspace Theorem allows us often to strengthen or to complement results established by means of the qualitative Schmidt Subspace Theorem; see for instance the survey [16]. In particular, by combining ideas from [6,7,8] with new arguments, we have obtained in [17] transcendence measures for transcendental real numbers whose sequence of partial quotients a is such that n → p(n, a)/n is bounded.…”

Section: Discussionmentioning

confidence: 99%

Automatic continued fractions are transcendental or quadratic

Bugeaud¹

2013

Ann. Sci. École Norm. Sup.

Self Cite

View full text Add to dashboard Cite

Abstract. We establish new combinatorial transcendence criteria for continued fraction expansions. Let α = [0; a 1 , a 2 , . . .] be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients (a ) ≥1 of α is not 'too simple' (in a suitable sense) and cannot be generated by a finite automaton. Résumé.Nousétablissons de nouveaux critères combinatoires de transcendance pour des développements en fraction continue. Soit α = [0; a 1 , a 2 , . . .] un nombre algébrique de degré au moinségalà trois. L'un de nos critères entraîne que la suite (a ) ≥1 des quotients partiels de α n'est pas trop simple (en un certain sens) et ne peut pasêtre engendrée par un automate fini.

show abstract

“…As shown in [4], this powerful criterion has many applications and yields among other things the transcendence of irrational real numbers whose expansion in some integer base can be generated by a finite automaton. The latter result was generalized in [8], where we gave transcendence measures for a large class of real numbers shown to be transcendental in [4]. The key ingredient for the proof is then the Quantitative Subspace Theorem [32].…”

Section: Introductionmentioning

confidence: 92%

“…If α has unbounded partial quotients, the key auxiliary ingredient is Proposition 11.1 of [8] Consequently, if α has unbounded partial quotients, then ξ is a U 2 -number.…”

Section: Proofs Of Theorems 221 and 222mentioning

confidence: 99%

Transcendence measures for continued fractions involving repetitive or symmetric patterns

Adamczewski

Bugeaud²

2010

J. Eur. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. There is a long tradition in constructing explicit classes of transcendental continued fractions and especially transcendental continued fractions with bounded partial quotients. By means of the Schmidt Subspace Theorem, existing results were recently substantially improved by the authors in a series of papers, providing new classes of transcendental continued fractions. It is the purpose of the present work to show how the Quantitative Subspace Theorem yields transcendence measures for (most of) these numbers.

show abstract

Nombres réels de complexité sous-linéaire : mesures d'irrationalité et de transcendance

Abstract: International audienc

Cited by 22 publications

References 35 publications

A new complexity function, repetitions in Sturmian words, and irrationality exponents of Sturmian numbers

A new complexity function, repetitions in Sturmian words, and irrationality exponents of Sturmian numbers

Automatic continued fractions are transcendental or quadratic

Transcendence measures for continued fractions involving repetitive or symmetric patterns

Contact Info

Product

Resources

About