9 research outputs found
On the critical exponent of generalized Thue-Morse words
For certain generalized Thue-Morse words t, we compute the "critical
exponent", i.e., the supremum of the set of rational numbers that are exponents
of powers in t, and determine exactly the occurrences of powers realizing it.Comment: 13 pages; to appear in Discrete Mathematics and Theoretical Computer
Science (accepted October 15, 2007
Arithmetical subword complexity of automatic sequences
We fully classify automatic sequences over a finite alphabet
with the property that each word over appears is along an
arithmetic progression. Using the terminology introduced by Avgustinovich,
Fon-Der-Flaass and Frid, these are the automatic sequences with the maximal
possible arithmetical subword complexity. More generally, we obtain an
asymptotic formula for arithmetical (and even polynomial) subword complexity of
a given automatic sequence .Comment: 14 pages, comments welcom
On the critical exponent of generalized Thue-Morse words
Automata, Logic and Semantic
Sequences of linear arithmetical complexity
AbstractArithmetical complexity of infinite sequences is the number of all words of a given length whose symbols occur in the sequence at positions which constitute arithmetical progressions. We show that uniformly recurrent sequences whose arithmetical complexity grows linearly are precisely Toeplitz words of a specific form
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Monochromatic arithmetic progressions in automatic sequences with group structure
We determine asymptotic growth rates for lengths of monochromatic arithmetic progressions in certain automatic sequences. In particular, we look at (one-sided) fixed points of aperiodic, primitive, bijective substitutions and spin substitutions, which are generalisations of the Thue-Morse and Rudin-Shapiro substitutions, respectively. For such infinite words, we show that there exists a subsequence {dn} of differences along which the maximum length A (dn) of a monochromatic arithmetic progression (with fixed difference dn) grows at least polynomially in dn. Explicit upper and lower bounds for the growth exponent can be derived from a finite group associated to the substitution. As an application, we obtain bounds for a van der Waerden-type number for a class of colourings parametrised by the size of the alphabet and the length of the substitution
Arithmetical complexity of symmetric D0L words
We characterize and count words which occur in arithmetical subsequences of xed points of symmetric morphisms
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Forward Limit Sets of Semigroups of Substitutions and Arithmetic Progressions in Automatic Sequences
This thesis deals with symbolic sequences generated by semigroups of substitutions acting on finite alphabets.
First, we investigate the underlying structure of certain automatic sequences by studying the maximum length A(d) of the monochromatic arithmetic progressions of difference d appearing in these sequences. For example, for the Thue-Morse sequence and a class of generalised Thue-Morse sequences, we give exact values of A(d) or upper bounds on it, for certain differences d. For aperiodic, primitive, bijective substitutions and spin substitutions, which are generalisations of the Thue-Morse and Rudin-Shapiro substitutions, respectively, we study the asymptotic growth rate of A(d). In particular, we prove that there exists a subsequence (d_n) of differences along which A(d_n) grows at least polynomially in d_n. Explicit upper and lower bounds for the growth exponent can be derived from a finite group associated to the substitution considered.
Next, we introduce the forward limit set Λ of a semigroup S generated by a family of substitutions of a finite alphabet, which typically coincides with the set of all possible s-adic limits of that family. We provide several alternative characterisations of the forward limit set. For instance, we prove that Λ is the unique maximal closed and strongly S-invariant subset of the space of all infinite words, and we prove that it is the closure of the image under S of the set of all fixed points of S. It is usually difficult to compute a forward limit set explicitly; however, we show that, provided certain assumptions hold, Λ is uncountable, and we supply upper bounds on its size in terms of logarithmic Hausdorff dimension