Related papers: Words and morphisms with Sturmian erasures

A word is said to be \emph{bordered} if it contains a non-empty proper prefix that is also a suffix. We can naturally extend this definition to pairs of non-empty words. A pair of words $(u,v)$ is said to be \emph{mutually bordered} if…

For which unary predicates $P_1, \ldots, P_m$ is the MSO theory of the structure $\langle \mathbb{N}; <, P_1, \ldots, P_m \rangle$ decidable? We survey the state of the art, leading us to investigate combinatorial properties of…

Let $r$ and $s$ be multiplicatively independent positive integers. We establish that the $r$-ary expansion and the $s$-ary expansion of an irrational real number, viewed as infinite words on $\{0, 1, \ldots , r-1\}$ and $\{0, 1, \ldots ,…

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…

We count the number of occurrences of certain patterns in given words. We choose these words to be the set of all finite approximations of a sequence generated by a morphism with certain restrictions. The patterns in our considerations are…

We implement a decision procedure for answering questions about a class of infinite words that might be called (for lack of a better name) "Fibonacci-automatic". This class includes, for example, the famous Fibonacci word f = 01001010...,…

We consider the structures given by repeatedly generalising the definition of finite state automata by symmetry considerations, and constructing analogues of transition monoids at each step. This approach first gives us non-deterministic…

We introduce and study a new complexity function in combinatorics on words, which takes into account the smallest second occurrence time of a factor of an infinite word. We characterize the eventually periodic words and the Sturmian words…

Given a finite alphabet X and an ordering on the letters, the map \sigma sends each monomial on X to the word that is the ordered product of the letter powers in the monomial. Motivated by a question on Groebner bases, we characterize…

In this paper, we analyze the density of the Fibonacci word and its derived forms by examining the morphisms associated with each. It offers a comparative analysis of the density of Fibonacci numbers alongside other words derived from…

It is known that each word of length $n$ contains at most $n+1$ distinct palindromes. A finite rich word is a word with maximal number of palindromic factors. The definition of palindromic richness can be naturally extended to infinite…

Let us consider an infinite word and $k\geq 1$ an integer. By steps of $k$, we substitute a letter ofthis infinite word by the power of an external letter. The new word obtaining by this process is called $k$ to $k$ substitution of a power…

A shuffle square is a word consisting of two shuffled copies of the same word. For instance, the Turkish word $\mathtt{\color{red}{ik}\color{blue}{i}\color{red}{li}\color{blue}{kli}}$ (binary in English) is a shuffle square, as it can be…

A set X of partial words over a finite alphabet A is called unavoidable if every two-sided infinite word over A has a factor compatible with an element of X. Unlike the case of a set of words without holes, the problem of deciding whether…

We show that two-dimensional billiard systems are Turing complete, in the sense that the halting of any Turing machine with a given input is equivalent to a certain bounded trajectory in this system entering a specified open set. Billiards…

We give a combinatorial definition of the second parameter describing Sturmian words, that we call formal intercept, as an infinite Ostrowski expansion. We recall the well-known properties of Sturmian words, study factors graphs and…

We construct an extension $E(A,G)$ of a given group $G$ by infinite non-Archimedean words over an discretely ordered abelian group like $Z^n$. This yields an effective and uniform method to study various groups that "behave like $G$". We…

A word over an ordered alphabet is said to be clustering if identical letters appear adjacently in its Burrows-Wheeler transform. Such words are strictly related to (discrete) interval exchange transformations. We use an extended version of…

Given a positive integer $N$ and $x$ irrational between zero and one, an $N$-continued fraction expansion of $x$ is defined analogously to the classical continued fraction expansion, but with the numerators being all equal to $N$. Inspired…

In the literature two notions of the word problem for a variety occur. A variety has a decidable word problem if every finitely presented algebra in the variety has a decidable word problem. It has a uniformly decidable word problem if…