An explicit lower bound for the block complexity of an algebraic number

Bugeaud, Yann

doi:10.4171/rlm/521

Cited by 4 publications

(5 citation statements)

References 8 publications

(11 reference statements)

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“…Furthermore, proceeding as in [15] and in [18], it seems to be possible to prove that if a = a 1 a 2 . .…”

Section: Discussionmentioning

confidence: 99%

Automatic continued fractions are transcendental or quadratic

Bugeaud¹

2013

Ann. Sci. École Norm. Sup.

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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.

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“…Furthermore, proceeding as in [15] and in [18], it seems to be possible to prove that if a = a 1 a 2 . .…”

Section: Discussionmentioning

confidence: 99%

Automatic continued fractions are transcendental or quadratic

Bugeaud¹

2013

Ann. Sci. École Norm. Sup.

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“…Let us denote respectively by d and H the degree and the height of α. Then the main result of [19] allows to extract the following upper bound : I(α, M ) ≤ max (max(log H, e)100kM 2 ) 8 log 4kM 2 , (log d)10 100 (kM 2 ) 11/2 log(kM 2 ) 2.1 . 6.3.…”

Section: 12mentioning

confidence: 99%

On the computational complexity of algebraic numbers: the Hartmanis–Stearns problem revisited

Adamczewski

Cassaigne

Gonidec³

2020

Trans. Amer. Math. Soc.

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We consider the complexity of integer base expansions of algebraic irrational numbers from a computational point of view. A major contribution in this area is that the base-b expansion of algebraic irrational real numbers cannot be generated by finite automata. Our aim is to provide two natural generalizations of this theorem. Our main result is that the base-b expansion of algebraic irrational real numbers cannot be generated by deterministic pushdown automata. Incidentally, this completely solves the Hartmanis-Stearns problem for the class of multistack machines. We also confirm an old claim of Cobham from 1968 proving that such real numbers cannot be generated by tag machines with dilation factor larger than one. In order to stick with the modern terminology, we also show that the latter generate the same class of real numbers than morphisms with exponential growth.1. This means computable in O(n log 1+ε n) operations for some ε. 2. As observed in [26], this may be the less surprising issue.

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“…Using R n 2, 186n 9 2 2n 50 2n , δ −2 log(3δ −1 R) (δ −1 log 3R) 3 , this leads to m 6m * × log 186n 9 2 2n δ −2 log(3δ −1 R) log(δ −1 log 3R) 6m * 2n log 50 log log 6 + 3 100nm * 10 5 2 2n n 10 δ −2 log(3δ −1 R),…”

Section: Proof Of Theorem 23mentioning

confidence: 99%

“…As it turned out, this version is in general more useful for applications than the existing quantitative versions of the basic Subspace Theorem concerning (1.1). For instance, the work of Evertse and Schlickewei led to uniform upper bounds for the number of solutions of linear equations in unknowns from a multiplicative group of finite rank [12] and for the zero multiplicity of linear recurrence sequences [27], and more recently to results on the complexity of b-ary expansions of algebraic numbers [6], [3], to improvements and generalizations of the Cugiani-Mahler theorem [2], and approximation to algebraic numbers by algebraic numbers [5]. For an overview of recent applications of the Quantitative Subspace Theorem we refer to Bugeaud's survey paper [4].…”

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confidence: 99%

A further improvement of the Quantitative Subspace Theorem

Evertse

Ferretti

2013

Ann. Math.

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In 2002, Evertse and Schlickewei [11] obtained a quantitative version of the so-called Absolute Parametric Subspace Theorem. This result deals with a parametrized class of twisted heights. One of the consequences of this result is a quantitative version of the Absolute Subspace Theorem, giving an explicit upper bound for the number of subspaces containing the solutions of the Diophantine inequality under consideration.In the present paper, we further improve Evertse's and Schlickewei's quantitative version of the Absolute Parametric Subspace Theorem, and deduce an improved quantitative version of the Absolute Subspace Theorem. We combine ideas from the proof of Evertse and Schlickewei (which is basically a substantial refinement of Schmidt's proof of his Subspace Theorem from 1972 [22]), with ideas from Faltings' and Wüstholz' proof of the Subspace Theorem [14].

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An explicit lower bound for the block complexity of an algebraic number

Abstract: Abstract. Let b ≥ 2 be an integer and ξ be an irrational real number. Among other results, we establish an explicit lower bound for the number of distinct blocks of n digits occurring in the b-ary expansion of ξ.

Cited by 4 publications

References 8 publications

Automatic continued fractions are transcendental or quadratic

Automatic continued fractions are transcendental or quadratic

On the computational complexity of algebraic numbers: the Hartmanis–Stearns problem revisited

A further improvement of the Quantitative Subspace Theorem

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