There are exponentially many ternary words that avoid abelian cubes

用二元子立方体覆盖三元立方体

Covering the ternary cube by binary subcubes

该研究针对用二元子立方体覆盖三元立方体的问题，确定了最小覆盖数f(n)的上界，解答了Imre Leader的问题，还得出f(n)与(3/2)ⁿ比值的单调性及相关常数范围。

On linear ternary Intersection sequences and their properties

Let $D^+$ be the first octant of the Euclidean space and consider the integral cube grid $G$ in $D^+$. The intersections of each line with $G$ form an infinite sequence of three letters which can be considered as an extension of well-known Sturmian words. A classification of such linear ternary sequences is presented and a family of examples is constructed from a notable sequence $S^M$ which could be viewed as an analogue of the Fibonacci word in the family of Sturmian words. The factor complexity and the palindromic complexity of these linear ternary sequences are also studied. The last resul…

Observations and Problems on k-abelian avoidability

We introduce new avoidability problems for words by considering equivalence relations, k-abelian equivalences, which lie properly in between equality and commutative equality, i.e. abelian equality. For two k-abelian equivalent words the numbers of occurrences of different factors of length k coincide and the prefixies (resp. suffixies) of length k-1 are equal as well. The size of the smallest alphabet avoiding 2-repetitions of words, i.e. squares, is three and for abelian squares it is four. It follows that for 2-abelian squares this size has to be three or four. Similarly, the size of the sm…

Avoidability of long $k$-abelian repetitions

We study the avoidability of long $k$-abelian-squares and $k$-abelian-cubes on binary and ternary alphabets. For $k=1$, these are Mäkelä's questions. We show that one cannot avoid abelian-cubes of abelian period at least $2$ in infinite binary words, and therefore answering negatively one question from Mäkelä. Then we show that one can avoid $3$-abelian-squares of period at least $3$ in infinite binary words and $2$-abelian-squares of period at least 2 in infinite ternary words. Finally we study the minimum number of distinct $k$-abelian-squares that must appear in an infinite binary word.

Cubefree words with many squares

We construct infinite cubefree binary words containing exponentially many distinct squares of length n. We also show that for every positive integer n, there is a cubefree binary square of length 2n.

Square-free reducts of words

A \emph{square} is a finite non-empty word consisting of two identical adjacent blocks. A word is \emph{square-free} if it does not contain a square as a factor. In any finite word one may delete the repeated block of a square, obtaining thereby a shorter word. By repeating this process, a square-free word is eventually reached, which we call a \emph{reduct} of the original word. How many different reducts a single word may have? It is not hard to prove that any binary word has exactly one reduct. We prove that there exist ternary words with arbitrarily many reducts. Moreover, the function cou…