We compare two sets of (infinite) binary sequences whose suffixes satisfy extremal conditions: one occurs when studying iterations of unimodal continuous maps from the unit interval into itself, but it also characterizes univoque real numbers; the other is a disguised version of the set of characteristic sturmian sequences. As a corollary to our study we obtain that a real number $\beta$ in $(1,2)$ is univoque and self-sturmian if and only if the $\beta$-expansion of $1$ is of the form $1v$, where $v$ is a characteristic...

We study the functional equation: $\left(1\right)ABC=CDA$ where $A,B,C$ and $D$ are words over an alphabet $\mathcal{A}$. In particular we prove a «structure result» for the inner factors $B,D$: for suitably chosen words $X,Y,Z$ one has: $\left(2\right)B=XYZ$, $D=ZYX$ $\left(2\right)B=XYZ$, $D=ZYX$ $\left(2\right)B=XYZ$, $D=ZYX$ $\left(2\right)B=XYZ$, $D=ZYX$. It is a generalization of the Lyndon-Schützenberger's Theorem (see [7]): if in (1) $A$ or $C$ is empty, formula (2) holds true with one among $X,Y,Z$ which can be chosen empty.

We give analogs of the complexity $p\left(n\right)$ and of Sturmian words which are called respectively the $*$-complexity $p_{*}\left(n\right)$ and $*$-Sturmian words. We show that the class of $*$-Sturmian words coincides with the class of words satisfying $p_{*}\left(n\right)\le n+1$, and we determine the structure of $*$-Sturmian words. For a class of words satisfying $p_{*}\left(n\right)=n+1$, we give a general formula and an upper bound for $p\left(n\right)$. Using this general formula, we give explicit formulae for $p\left(n\right)$ for some words belonging to this class. In general, $p\left(n\right)$ can take large...

Let $k\ge 2$ and define $F^{\left(k\right)}:={\left(F_n^{\left(k\right)}\right)}_{n\ge 0}$, the $k$-generalized Fibonacci sequence whose terms satisfy the recurrence relation $F_n^{\left(k\right)}=F_{n-1}^{\left(k\right)}+F_{n-2}^{\left(k\right)}+\cdots +F_{n-k}^{\left(k\right)}$, with initial conditions $0,0,\cdots ,0,1$ ($k$ terms) and such that the first nonzero term is $F_1^{\left(k\right)}=1$. The sequences $F:=F^{\left(2\right)}$ and $T:=F^{\left(3\right)}$ are the known Fibonacci and Tribonacci sequences, respectively. In 2005, Noe and Post made a conjecture related to the possible solutions of the Diophantine equation $F_n^{\left(k\right)}=F_m^{\left(\ell \right)}$. In this note, we use transcendental tools to provide a general method for finding the intersections $F^{\left(k\right)}\cap F^{\left(m\right)}$ which gives...

Integer sequences of the form $\lfloor n^c\rfloor$, where 1 < c < 2, can be locally approximated by sequences of the form ⌊nα+β⌋ in a very good way. Following this approach, we are led to an estimate of the difference ${\sum }_{n\le x}\phi \left(\lfloor n^c\rfloor \right)-1/c{\sum }_{n\le x^c}\phi \left(n\right)n^{1/c-1}$, which measures the deviation of the mean value of φ on the subsequence $\lfloor n^c\rfloor$ from the expected value, by an expression involving exponential sums. As an application we prove that for 1 < c ≤ 1.42 the subsequence of the Thue-Morse sequence indexed by $\lfloor n^c\rfloor$ attains both of its values with...