ИСТИНА

The main goal of this presentation is to present and apply the notions of complexity function to some directly connected properties of the words. This work is strongly related to the courses Combinatorics on Words and Symbolic Dynamics that I followed. Let \textbf{w} be a (possibly infinite) word of symbols from an alphabet. In computer science, the complexity function of a word is the function $p_{\textbf{w}}(n)$ of a positive integer $n$ to be the number of different factors (consecutive subwords) of length $n$ occurring in the word \textbf{w}.Then we study the complexity of Sturmian words which are aperiodic words with the least factor complexity and show a well-known characterization of Sturmian words as codings of rotations with an irrational angle and a convenient partition of the perimeter of the circle. Making use of that alternative definition, we are able to show that, for all $n > 0$, the word made by the left-most (most significant digit) of the sequence $2^n$, $\textbf{w} = 248136125...$ can be described by these words (otherwise stated the $j^{th}$ digit occurring in $\textbf{w}$ is the left-most digit in the decimal expansion of $2^j$). The next section of this presentation explains that the sequence made by powers of $2$ one the circle with irrational angle is dense. I prove in that case that $p_{\textbf{w}}(n)$. Finally, the last section is dedicated to the very interesting relation existing between k-dimensional torus with the sequence of left-most digit occurring in the decimal representation of $2$ to the power $n^k$, where $n, k$ are positive integers. The main result of our work is: Theorem. Given a natural $k$. The word $\textbf{w}_n$ is considered, $n^{th}$ term of which is the first digit of the number $2^{n^k}$ in decimal representation. Then there is a polynomial $P(k)$ of degree $k(k+1)/2$ such that in this sequence the number of distinct \textit{words}; of length $k$ for sufficiently large $k$ is equal to $P(k)$.