Results 1  10
of
22
Initial powers of Sturmian sequences
 Acta Arith
, 2006
"... Abstract. In this paper we investigate powers of prefixes of Sturmian sequences. We give an explicit formula for ice(ω), the initial critical exponent of a Sturmian sequence ω, defined as the supremum of all real numbers p>0 for which there exist arbitrary long prefixes of ω of the form u p, in term ..."
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Abstract. In this paper we investigate powers of prefixes of Sturmian sequences. We give an explicit formula for ice(ω), the initial critical exponent of a Sturmian sequence ω, defined as the supremum of all real numbers p>0 for which there exist arbitrary long prefixes of ω of the form u p, in terms of its Sadic representation. This formula is based on Ostrowski’s numeration system. Furthermore we characterize those irrational slopes α of which there exists a Sturmian sequence ω beginning in only finitely many powers of 2 + ε, that is for which ice(ω) =2. In the process we recover the known results for the index (or critical exponent) of a Sturmian sequence. We also focus on the Fibonacci Sturmian shift and prove that the set of Sturmian sequences with ice strictly smaller than its everywhere value has
REVERSALS AND PALINDROMES IN CONTINUED FRACTIONS
"... Abstract. Several results on continued fractions expansions are direct on indirect consequences of the mirror formula. We survey occurrences of this formula for Sturmian real numbers, for (simultaneous) Diophantine approximation, and for formal power series. 1. ..."
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Cited by 8 (4 self)
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Abstract. Several results on continued fractions expansions are direct on indirect consequences of the mirror formula. We survey occurrences of this formula for Sturmian real numbers, for (simultaneous) Diophantine approximation, and for formal power series. 1.
Initial Powers Of Sturmian Words
 Acta Arith
, 2001
"... We study powers of prefixes of Sturmian words. 1. ..."
A PROOF OF DEJEAN’S CONJECTURE
, 905
"... Abstract. We prove Dejean’s conjecture. Specifically, we show that Dejean’s conjecture holds for the last remaining open values of n, namely 15 ≤ n ≤ 26. 1. ..."
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Cited by 3 (0 self)
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Abstract. We prove Dejean’s conjecture. Specifically, we show that Dejean’s conjecture holds for the last remaining open values of n, namely 15 ≤ n ≤ 26. 1.
Some Properties of the singular words of the Fibonacci word
, 1994
"... The combinatorial properties of the Fibonacci infinite word are of great interest in some aspects of mathematics and physics, such as number theory, fractal geometry, formal language, computational complexity, quasicrystals etc. In this note, we introduce the singular words of the Fibonacci infinite ..."
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The combinatorial properties of the Fibonacci infinite word are of great interest in some aspects of mathematics and physics, such as number theory, fractal geometry, formal language, computational complexity, quasicrystals etc. In this note, we introduce the singular words of the Fibonacci infinite word and discuss their properties. We establish two decompositions of the Fibonacci word in singular words and their consequences. By using these results, we discuss the local isomorphism of the Fibonacci word and the overlap properties of the factors. Moreover, we also give new proofs for the results on special words and the power of the factors. The combinatorial properties of the Fibonacci infinite word are of great interest in some aspects of mathematics and physics, such as number theory, fractal geometry, formal language, computational complexity, quasicrystals etc. See [1, 3, 8, 9, 11]. Moreover, the properties of the subwords of the Fibonacci infinite word have been studied extensiv...
Dejean’s conjecture holds for n ≥ 27
, 2009
"... We show that Dejean’s conjecture holds for n ≥ 27. Repetitions in words have been studied since the beginning of the previous century [13, 14]. Recently, there has been much interest in repetitions with fractional exponent [1, 3, 5, 6, 7, 9]. For rational 1 < r ≤ 2, a fractional rpower is a nonemp ..."
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We show that Dejean’s conjecture holds for n ≥ 27. Repetitions in words have been studied since the beginning of the previous century [13, 14]. Recently, there has been much interest in repetitions with fractional exponent [1, 3, 5, 6, 7, 9]. For rational 1 < r ≤ 2, a fractional rpower is a nonempty word w = xx ′ such that x ′ is the prefix of x of length (r − 1)x. For example, 010 is a 3/2power. A basic problem is that of identifying the repetitive threshold for each alphabet size n> 1: What is the infimum of r such that an infinite sequence on n letters exists, not containing any factor of exponent greater than r? The infimum is called the repetitive threshold of an nletter alphabet, denoted by RT(n). Dejean’s conjecture [5] is that ⎨ 7/4, n = 3 RT(n) = 7/5, n = 4 n/(n − 1) n ̸ = 3, 4 Thue, Dejean and Pansiot, respectively [14, 5, 12] established the values RT(2), RT(3), RT(4). MoulinOllagnier [11] verified Dejean’s conjecture for 5 ≤ n ≤ 11, and MohammadNoori and Currie [10] proved the conjecture for 12 ≤ n ≤ 14.
E.: Relation between powers of factors and recurrence function characterizing Sturmian words, submitted, arXiv 0809.0603v2[math.CO
"... In this paper we use the relation of the index of an infinite aperiodic word and its recurrence function to give another characterization of Sturmian words. As a byproduct, we give a new proof of theorem describing the index of a Sturmian word in terms of the continued fraction expansion of its slop ..."
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In this paper we use the relation of the index of an infinite aperiodic word and its recurrence function to give another characterization of Sturmian words. As a byproduct, we give a new proof of theorem describing the index of a Sturmian word in terms of the continued fraction expansion of its slope. This theorem was independently proved in [7] and [9]. 1
Locally Periodic Infinite Words and a Chaotic Behaviour
"... We call a oneway infinite word w over a finite alphabet (ae; p)repetitive if all long enough prefixes of w contain as a suffix a aeth power (or more generally a repetition of order ae) of a word of length at most p. We show that each (2; 4)repetitive word is ultimately periodic, as well as that ..."
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We call a oneway infinite word w over a finite alphabet (ae; p)repetitive if all long enough prefixes of w contain as a suffix a aeth power (or more generally a repetition of order ae) of a word of length at most p. We show that each (2; 4)repetitive word is ultimately periodic, as well as that there exist nondenumerably many, and hence also nonultimately periodic, (2; 5) repetitive words. Further we characterize nonultimately periodic (2; 5) repetitive words both structurally and algebraically. Supported by Academy of Finland under the grant 14047. y On leave from Instytut Informatyli UW, Banacha 2, 02047 Warszawa, Poland. 1 Introduction One of the fundamental topics in mathematical research is to search for connections between local and global regularities. We consider such a problem in connection with infinite words. The regularity is specified as a periodicity. Our research is motivated by a remarkable result of Mignosi, Restivo and Salemi (cf. [MRS]) where they chara...