Kalle Saari

Abelian complexity of minimal subshifts

In this paper we undertake the general study of the Abelian complexity of an infinite word on a finite alphabet. We investigate both similarities and differences between the Abelian complexity and the usual subword complexity. While the Thue-Morse minimal subshift is neither characterized by its Abelian complexity nor by its subword complexity alone, we show that the subshift is completely characterized by the two complexity functions together. We give an affirmative answer to an old question of Rauzy by exhibiting a class of words whose Abelian complexity is everywhere equal to 3. We also investigate links between Abelian complexity and the existence of Abelian powers. Using van der Waerdens theorem, we show that any minimal subshift having bounded Abelian complexity contains Abelian k-powers for every positive integer k. In the case of Sturmian words, we prove something stronger: for every Sturmian word ω and positive integer k, each sufficiently long factor of ω begins with an Abelian k-power.

AVOIDING ABELIAN POWERS IN BINARY WORDS WITH BOUNDED ABELIAN COMPLEXITY

The notion of Abelian complexity of infinite words was recently used by the three last authors to investigate various Abelian properties of words. In particular, using van der Waerdens theorem, they proved that if a word avoids Abelian k-powers for some integer k, then its Abelian complexity is unbounded. This suggests the following question: How frequently do Abelian k-powers occur in a word having bounded Abelian complexity? In particular, does every uniformly recurrent word having bounded Abelian complexity begin in an Abelian k-power? While this is true for various classes of uniformly recurrent words, including for example the class of all Sturmian words, in this paper we show the existence of uniformly recurrent binary words, having bounded Abelian complexity, which admit an infinite number of suffixes which do not begin in an Abelian square. We also show that the shift orbit closure of any infinite binary overlap-free word contains a word which avoids Abelian cubes in the beginning. We also consider the effect of morphisms on Abelian complexity and show that the morphic image of a word having bounded Abelian complexity has bounded Abelian complexity. Finally, we give an open problem on avoidability of Abelian squares in infinite binary words and show that it is equivalent to a well-known open problem of Pirillo–Varricchio and Halbeisen–Hungerbuhler.

LEAST PERIODS OF FACTORS OF INFINITE WORDS

Work of the first author supported by a Discovery Grant from NSERC. Work of the second author supported by the Finnish Academy under grant 8206039.

Local squares, periodicity and finite automata

We consider the general problem when local regularity implies the global one in the setting where local regularity means the existence of a square of certain length in every position of an infinite word. The square can occur as centered or to the left or to the right from each position. In each case there are three variants of the problem depending on whether the square is that of words, that of abelian words or, as an in between case, that of so called k-abelian words. The above nine variants of the problem are completely solved, and some open problems are addressed in the k-abelian case. Finally, an amazing unavoidability result for 2-abelian squares is obtained.

On the frequency of letters in morphic sequences

A necessary and sufficient criterion for the existence and value of the frequency of a letter in a morphic sequence is given. This is done using a certain incidence matrix associated with the morphic sequence. The characterization gives rise to a simple if-and-only-if condition that all letter frequencies exist.

Primitive Words and Lyndon Words in Automatic and Linearly Recurrent Sequences

We investigate questions related to the presence of primitive words and Lyndon words in automatic and linearly recurrent sequences. We show that the Lyndon factorization of a k-automatic sequence is itself k-automatic. We also show that the function counting the number of primitive factors (resp., Lyndon factors) of length n in a k-automatic sequence is k-regular. Finally, we show that the number of Lyndon factors of a linearly recurrent sequence is bounded.

On highly palindromic words

We study some properties of palindromic (scattered) subwords of binary words. In view of the classical problem on subwords, we show that the set of palindromic subwords of a word characterizes the word up to reversal. Since each word trivially contains a palindromic subword of length at least half of its length-a power of the prevalent letter-we call a word that does not contain any palindromic subword longer than half of its length minimal palindromic. We show that every minimal palindromic word is abelian unbordered, that is, no proper suffix of the word can be obtained by permuting the letters of a proper prefix. We also propose to measure the degree of palindromicity of a word w by the ratio |rws|/|w|, where the word rws is minimal palindromic and rs is as short as possible. We prove that the ratio is always bounded by four, and construct a sequence of words that achieves this bound asymptotically.

Everywhere α-repetitive sequences and Sturmian words

Local constraints on an infinite sequence that imply global regularity are of general interest in combinatorics on words. We consider this topic by studying everywhere α-repetitive sequences, sequences in which every position has an occurrence of a repetition of order α ≥ 1 of bounded length. The number of minimal such repetitions, called minimal α-powers, is then finite. A natural question regarding global regularity is to determine the least number of minimal α-powers such that an α-repetitive sequence is not necessarily ultimately periodic. We solve this question for 1 ≤ α 17/8. We also show that Sturmian words are among the optimal 2 - and 2+-repetitive sequences.

On the frequency of letters in pure binary morphic sequences

It is well-known that the frequency of letters in primitive morphic sequences exists. We show that the frequency of letters exists in pure binary morphic sequences generated by non-primitive morphisms. Therefore, the letter frequency exists in every pure binary morphic sequence. We also show that this is somewhat optimal, in the sense that the result does not hold in the class of general binary morphic sequences. Finally, we give an explicit formula for the frequency of letters.

Extremal Words in the Shift Orbit Closure of a Morphic Sequence

Given an infinite word \(\ensuremath{\mathbf{x}} \) over an alphabet A, a letter b occurring in \(\ensuremath{\mathbf{x}} \), and a total order σ on A, we call the smallest word with respect to σ starting with b in the shift orbit closure \(\ensuremath{\mathcal{S}} _{\ensuremath{\mathbf{x}} }\) of \(\ensuremath{\mathbf{x}} \) an extremal word of \(\ensuremath{\mathbf{x}} \). In this paper we consider the extremal words of morphic words. If \(\ensuremath{\mathbf{x}} = g(f^{\omega}(a))\) for some morphisms f and g, we give a simple condition on f and g that guarantees that all extremal words are morphic. An application of this condition shows that all extremal words of binary pure morphic words are morphic. Our technique also yields easy characterizations of extremal words of the Period-doubling and Chacon words and a new proof of the form of the lexicographically least word in the shift orbit closure of the Rudin-Shapiro word.