On the complexity of algebraic numbers I. Expansions in integer bases

Adamczewski, Boris; Bugeaud, Yann

doi:10.4007/annals.2007.165.547

Cited by 140 publications

(215 citation statements)

References 38 publications

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“…So far the equality P(α, n) = q n is out of reach. By a result of Adamczewski and Bugeaud [1], we know that P(α, n)/n → ∞ as n → ∞ for each algebraic irrational number α. One among earlier results [7] implies that P(α, n) − n → ∞ as n → ∞.…”

Section: Be a Sequence Of Positive Integers Given By X Nmentioning

confidence: 99%

On Integer Sequences Generated by Linear Maps

Dubickas¹

2009

Glasgow Math. J.

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Abstract. Let x 0 < x 1 < x 2 < · · · be an increasing sequence of positive integers given by the formula x n = βx n−1 + γ for n = 1, 2, 3, . . . , where β > 1 and γ are real numbers and x 0 is a positive integer. We describe the conditions on integers b d , . . . , b 0 , not all zero, and on a real number β > 1 under which the sequence of integers w n = b d x n+d + · · · + b 0 x n , n = 0, 1, 2, . . . , is bounded by a constant independent of n. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequence qx n+1

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Section: Be a Sequence Of Positive Integers Given By X Nmentioning

confidence: 99%

On Integer Sequences Generated by Linear Maps

Dubickas¹

2009

Glasgow Math. J.

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“…See, for instance, [1] for a recent progress on the distribution of digits in the expansions of algebraic irrational numbers in base b ≥ 2.…”

Section: Belongs To S By Theorem 2(ii) All Irreducible Polynomials Omentioning

confidence: 99%

Even and Odd Integral Parts of Powers of a Real Number

Dubickas¹

2006

Glasgow Math. J.

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Abstract. We define a subset Z of (1, +∞) with the property that for each α ∈ Z there is a nonzero real number ξ = ξ (α) such that the integral parts [ξα n ] are even for all n ∈ ‫.ގ‬ A result of Tijdeman implies that each number greater than or equal to 3 belongs to Z. However, Mahler's question on whether the number 3/2 belongs to Z or not remains open. We prove that the set S := (1, +∞) \ Z is nonempty and find explicitly some numbers in Z ∩ (5/4, 3) and in S ∩ (1, 2).2000 Mathematics Subject Classification. 11J71, 11R04, 11R06, 11R09. Introduction.Suppose that α > 1 is a real number. Then either for any real number ξ = 0 the fractional parts {ξα n } are greater than or equal to 1/2 for infinitely many n ∈ ‫ގ‬ or there exists a real number ξ = ξ (α) = 0 such that {ξα n } < 1/2 for every n ∈ ‫.ގ‬ The set of α > 1 for which the first possibility holds will be denoted by S. Similarly, we will denote by Z the set of α > 1 for which the second possibility holds, so S ∩ Z = ∅ and S ∪ Z = (1, +∞). Equivalently, α ∈ Z if and only if there is a real number ξ = ξ (α) = 0 such that the integral parts [ξα n ], n = 1, 2, . . . , are all even. In 1968, Mahler [16] asked whether the number 3/2 belongs to S or to Z (see also [20]). There are many reasons to believe that 3/2 ∈ S. Although for a 'random' pair ξ, α the fractional parts {ξα n }, n = 1, 2, . . . , are uniformly distributed in [0, 1) (see [13], [21]), the behavior of the sequence {ξα n }, n = 1, 2, . . . , for almost any 'specific' pair ξ, α, where Trivially, {2, 3, 4, . . .} ⊂ Z, because, for any integer g ≥ 2, by taking ξ = 2 we see that the numbers [2g n ] = 2g n , n = 1, 2, . . . , are all even. In general, one may expect that 'large' α belong to the set Z (see, for instance, Tijdeman's result [19] stated in Theorem 1(i) below) whereas 'small' α lie in S. In connection with this, one can ask whether there exist α > α > 1 such that α ∈ S and α ∈ Z. If the answer to this question were negative then the sets S and Z would simply be two intervals. Unfortunately, the situation is not that simple, because such α and α do exist. We will show, for instance, that the set Z contains a number smaller than 1.26 and that the set S contains the golden mean (1 + √ 5)/2 = 1.61803 . . . . In order to state our theorems, we recall first that α > 1 is called a Pisot number if it is an algebraic integer whose conjugates over ‫ޑ‬ different from α itself lie in the open unit disc. A Pisot number is called a strong Pisot number if it is not a rational integer and if its second largest (in modulus) conjugate is positive (see [2] and [6]). Also,

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“…Their combinatorial transcendence criterion was subsequently considerably improved in [10] by means of the multidimensional extension of Roth's theorem established by W. M. Schmidt, commonly referred to as the Schmidt Subspace Theorem [30,31]. 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].…”

Section: Introductionmentioning

confidence: 99%

“…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: 99%

Transcendence measures for continued fractions involving repetitive or symmetric patterns

Adamczewski

Bugeaud²

2010

J. Eur. Math. Soc.

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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.

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On the complexity of algebraic numbers I. Expansions in integer bases

Cited by 140 publications

References 38 publications

On Integer Sequences Generated by Linear Maps

On Integer Sequences Generated by Linear Maps

Even and Odd Integral Parts of Powers of a Real Number

Transcendence measures for continued fractions involving repetitive or symmetric patterns

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