Transcendence of Numbers with a Low Complexity Expansion

Ferenczi, Sébastien; Mauduit, Christian

doi:10.1006/jnth.1997.2175

Cited by 84 publications

(68 citation statements)

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“…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 → ∞. Analogously, in our problem, Theorem 3 implies that P(w, n) − n → ∞ as n → ∞ in case p < q 2 .…”

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

confidence: 76%

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 Nsupporting

confidence: 76%

On Integer Sequences Generated by Linear Maps

Dubickas¹

2009

Glasgow Math. J.

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“…By a result of Morse and Hedlund [18,19], every infinite word w that is not ultimately periodic satisfies p(n, w) ≥ n + 1 for n ≥ 1. Consequently, p(n, ξ, b) ≥ n + 1 for every positive integer n. This lower bound was subsequently improved upon in 1997 by Ferenczi and Mauduit [15], who applied a non-Archimedean extension of Roth's theorem established by Ridout [21] to show (see also [4]) that…”

Section: 09mentioning

confidence: 99%

On two notions of complexity of algebraic numbers

Bugeaud

Evertse

2008

Acta Arith.

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“…This condition distinguishes them from other sequences of complexity 2n 4-1 such as those obtained by coding trajectories of 3-interval exchange transformations [15], [16] or those of Chacon type, i.e., topologically isomorphic to the subshift generated by the Chacon sequence [8], [14]. A-R sequences have seen a recent surge of interest: [3], [7], [9], [10], [II], [17], [21], [22], [29], [30], [31]. In [I] Arnoux showed that the Tribonacci sequence may be geometrically realized by an exchange of six intervals on the circle.…”

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confidence: 99%