106 research outputs found

    Enumerating Abelian Returns to Prefixes of Sturmian Words

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    We follow the works of Puzynina and Zamboni, and Rigo et al. on abelian returns in Sturmian words. We determine the cardinality of the set APRu\mathcal{APR}_u of abelian returns of all prefixes of a Sturmian word uu in terms of the coefficients of the continued fraction of the slope, dependingly on the intercept. We provide a simple algorithm for finding the set APRu\mathcal{APR}_u and we determine it for the characteristic Sturmian words.Comment: 19page

    Abelian returns in Sturmian words

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    In this paper we study an abelian version of the notion of return word. Our main result is a new characterization of Sturmian words via abelian returns. Namely, we prove that a word is Sturmian if and only if each of its factors has two or three abelian returns. In addition, we describe the structure of abelian returns in Sturmian words, and discuss connections between abelian returns and periodicity

    Some properties of abelian return words (long abstract)

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    We investigate some properties of abelian return words as recently introduced by Puzynina and Zamboni. In particular, we obtain a characterization of Sturmian words with non-null intercept in terms of the finiteness of the set of abelian return words to all prefixes. We describe this set of abelian returns for the Fibonacci word but also for the 2-automatic Thue–Morse word. We also investigate the relationship existing between abelian complexity and finiteness of the set of abelian returns to all prefixes. We end this paper by considering the notion of abelian derived sequence. It turns out that, for the Thue–Morse word, the set of abelian derived sequences is infinite

    On a generalization of Abelian equivalence and complexity of infinite words

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    In this paper we introduce and study a family of complexity functions of infinite words indexed by k \in \ints ^+ \cup {+\infty}. Let k \in \ints ^+ \cup {+\infty} and AA be a finite non-empty set. Two finite words uu and vv in AA^* are said to be kk-Abelian equivalent if for all xAx\in A^* of length less than or equal to k,k, the number of occurrences of xx in uu is equal to the number of occurrences of xx in v.v. This defines a family of equivalence relations k\thicksim_k on A,A^*, bridging the gap between the usual notion of Abelian equivalence (when k=1k=1) and equality (when k=+).k=+\infty). We show that the number of kk-Abelian equivalence classes of words of length nn grows polynomially, although the degree is exponential in k.k. Given an infinite word \omega \in A^\nats, we consider the associated complexity function \mathcal {P}^{(k)}_\omega :\nats \rightarrow \nats which counts the number of kk-Abelian equivalence classes of factors of ω\omega of length n.n. We show that the complexity function P(k)\mathcal {P}^{(k)} is intimately linked with periodicity. More precisely we define an auxiliary function q^k: \nats \rightarrow \nats and show that if Pω(k)(n)<qk(n)\mathcal {P}^{(k)}_{\omega}(n)<q^k(n) for some k \in \ints ^+ \cup {+\infty} and n0,n\geq 0, the ω\omega is ultimately periodic. Moreover if ω\omega is aperiodic, then Pω(k)(n)=qk(n)\mathcal {P}^{(k)}_{\omega}(n)=q^k(n) if and only if ω\omega is Sturmian. We also study kk-Abelian complexity in connection with repetitions in words. Using Szemer\'edi's theorem, we show that if ω\omega has bounded kk-Abelian complexity, then for every D\subset \nats with positive upper density and for every positive integer N,N, there exists a kk-Abelian NN power occurring in ω\omega at some position $j\in D.

    Open and closed complexity of infinite words

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    In this paper we study the asymptotic behaviour of two relatively new complexity functions defined on infinite words and their relationship to periodicity. Given a factor ww of an infinite word x=x1x2x3x=x_1x_2x_3\cdots with each xix_i belonging to a fixed finite set A,\mathbb{A}, we say ww is closed if either wAw\in \mathbb{A} or if ww is a complete first return to some factor vv of x.x. Otherwise ww is said to be open. We show that for an aperiodic word xAN,x\in \mathbb{A}^\mathbb{N}, the complexity functions ClxCl_x (resp. Opx)Op_x) that count the number of closed (resp. open) factors of xx of each given length are both unbounded. More precisely, we show that if xx is aperiodic then lim infnNOpx(n)=+\liminf_{n\in \mathbb{N}} Op_x(n)=+\infty and lim supnSClx(n)=+\limsup_{n\in S} Cl_x(n)=+\infty for any syndetic subset SS of N.\mathbb{N}. However, there exist aperiodic infinite words xx verifying lim infnNClx(n)<+.\liminf_{n\in \mathbb{N}}Cl_x(n)<+\infty. Keywords: word complexity, periodicity, return words

    Approximate invariance for ergodic actions of amenable groups

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    We develop in this paper some general techniques to analyze action sets of small doubling for probability measure-preserving actions of amenable groups. As an application of these techniques, we prove a dynamical generalization of Kneser's celebrated density theorem for subsets in (\bZ,+), valid for any countable amenable group, and we show how it can be used to establish a plethora of new inverse product set theorems for upper and lower asymptotic densities. We provide several examples demonstrating that our results are optimal for the settings under study

    Privileged Words and Sturmian Words

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    This dissertation has two almost unrelated themes: privileged words and Sturmian words. Privileged words are a new class of words introduced recently. A word is privileged if it is a complete first return to a shorter privileged word, the shortest privileged words being letters and the empty word. Here we give and prove almost all results on privileged words known to date. On the other hand, the study of Sturmian words is a well-established topic in combinatorics on words. In this dissertation, we focus on questions concerning repetitions in Sturmian words, reproving old results and giving new ones, and on establishing completely new research directions. The study of privileged words presented in this dissertation aims to derive their basic properties and to answer basic questions regarding them. We explore a connection between privileged words and palindromes and seek out answers to questions on context-freeness, computability, and enumeration. It turns out that the language of privileged words is not context-free, but privileged words are recognizable by a linear-time algorithm. A lower bound on the number of binary privileged words of given length is proven. The main interest, however, lies in the privileged complexity functions of the Thue-Morse word and Sturmian words. We derive recurrences for computing the privileged complexity function of the Thue-Morse word, and we prove that Sturmian words are characterized by their privileged complexity function. As a slightly separate topic, we give an overview of a certain method of automated theorem-proving and show how it can be applied to study privileged factors of automatic words. The second part of this dissertation is devoted to Sturmian words. We extensively exploit the interpretation of Sturmian words as irrational rotation words. The essential tools are continued fractions and elementary, but powerful, results of Diophantine approximation theory. With these tools at our disposal, we reprove old results on powers occurring in Sturmian words with emphasis on the fractional index of a Sturmian word. Further, we consider abelian powers and abelian repetitions and characterize the maximum exponents of abelian powers with given period occurring in a Sturmian word in terms of the continued fraction expansion of its slope. We define the notion of abelian critical exponent for Sturmian words and explore its connection to the Lagrange spectrum of irrational numbers. The results obtained are often specialized for the Fibonacci word; for instance, we show that the minimum abelian period of a factor of the Fibonacci word is a Fibonacci number. In addition, we propose a completely new research topic: the square root map. We prove that the square root map preserves the language of any Sturmian word. Moreover, we construct a family of non-Sturmian optimal squareful words whose language the square root map also preserves.This construction yields examples of aperiodic infinite words whose square roots are periodic.Siirretty Doriast
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