On combinatorial properties of the Arshon sequence

AbstractWe consider combinatorial and algebraic properties of the language of factors of the infinite sequence on the three-letter alphabet built by S.E. Arshon in 1930s. This sequence never contains two successive equal words, i. e., avoids the number 2. The notion of avoidability is extended from integers to rational numbers. It is shown that the avoidability bound for the considered language is 74. This language is defined by two alternating morphisms; our method allows to study it like a formal language defined by one morphism. We also give a complete description of the syntactic congruence of the considered language

Concerning Kurosaki's squarefree word

In 2008, Kurosaki gave a new construction of a (bi-)infinite squarefree word over three letters. We show that in fact Kurosaki's word avoids 7/4+-powers, which, as shown by Dejean, is optimal over a 3-letter alphabet

The critical exponent of the Arshon words

Generalizing the results of Thue (for n = 2) and of Klepinin and Sukhanov (for n = 3), we prove that for all n greater than or equal to 2, the critical exponent of the Arshon word of order $n$ is given by (3n-2)/(2n-2), and this exponent is attained at position 1.Comment: 11 page

Critical Exponents and Stabilizers of Infinite Words

This thesis concerns infinite words over finite alphabets. It contributes to two topics in this area: critical exponents and stabilizers. Let w be a right-infinite word defined over a finite alphabet. The critical exponent of w is the supremum of the set of exponents r such that w contains an r-power as a subword. Most of the thesis (Chapters 3 through 7) is devoted to critical exponents. Chapter 3 is a survey of previous research on critical exponents and repetitions in morphic words. In Chapter 4 we prove that every real number greater than 1 is the critical exponent of some right-infinite word over some finite alphabet. Our proof is constructive. In Chapter 5 we characterize critical exponents of pure morphic words generated by uniform binary morphisms. We also give an explicit formula to compute these critical exponents, based on a well-defined prefix of the infinite word. In Chapter 6 we generalize our results to pure morphic words generated by non-erasing morphisms over any finite alphabet. We prove that critical exponents of such words are algebraic, of a degree bounded by the alphabet size. Under certain conditions, our proof implies an algorithm for computing the critical exponent. We demonstrate our method by computing the critical exponent of some families of infinite words. In particular, in Chapter 7 we compute the critical exponent of the Arshon word of order n for n ≥ 3. The stabilizer of an infinite word w defined over a finite alphabet Σ is the set of morphisms f: Σ*→Σ* that fix w. In Chapter 8 we study various problems related to stabilizers and their generators. We show that over a binary alphabet, there exist stabilizers with at least n generators for all n. Over a ternary alphabet, the monoid of morphisms generating a given infinite word by iteration can be infinitely generated, even when the word is generated by iterating an invertible primitive morphism. Stabilizers of strict epistandard words are cyclic when non-trivial, while stabilizers of ultimately strict epistandard words are always non-trivial. For this latter family of words, we give a characterization of stabilizer elements. We conclude with a list of open problems, including a new problem that has not been addressed yet: the D0L repetition threshold