Automatic Sequences: Theory, Applications, Generalizations by Jean-Paul Allouche

By Jean-Paul Allouche

Combining options of arithmetic and laptop technological know-how, this booklet is set the sequences of symbols that may be generated by means of basic versions of computation referred to as ''finite automata''. appropriate for graduate scholars or complicated undergraduates, it begins from hassle-free ideas and develops the fundamental concept. The learn then progresses to teach how those rules might be utilized to resolve difficulties in quantity idea and physics.

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T. Brown [1971] gave a survey on constructing strongly nonrepetitive sequences. Entringer, Jackson, and Schatz [1974] proved that every infinite word over a 2-letter alphabet contains arbitrarily long abelian squares. Ker¨anen [1992] solved Erd˝os’s problem by exhibiting a strongly nonrepetitive sequence over a 4-letter alphabet. Carpi [1998] showed that there are uncountably many abelian squarefree words over a 4-letter alphabet, and that the number of abelian squarefree words of each length grows exponentially.

Mignosi and Pirillo [1992] proved√that the critical exponent for the Fibonacci . 618. For other results on critical 2 exponents, see Klepinin and Sukhanov [1999], Vandeth [2000], and Damanik and Lenz [2002]. Erd˝os [1961, p. 240] first raised the problem of the existence of infinite abelian squarefree words. ) Evdokimov [1968] constructed such a sequence on 25 symbols. Pleasants [1970] improved this to 5 symbols. T. Brown [1971] gave a survey on constructing strongly nonrepetitive sequences. Entringer, Jackson, and Schatz [1974] proved that every infinite word over a 2-letter alphabet contains arbitrarily long abelian squares.

1 The critical exponent of the Thue–Morse word t is 2. Proof. The word t begins 011 · · · and hence contains a square. If t contained a (2 + )-power for any > 0, then it would contain an overlap. 1. There also exist various generalizations of squarefreeness. We say a word is an abelian square if it is of the form w w where w is a permutation of w. A word is abelian squarefree if it contains no abelian squares. 11) for more information. Another generalization is to study more general pattern avoidance problems.

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