Ľubomíra Balková

# Abstract:

We describe factor frequencies of the generalized Thue-Morse word ${\mathbf t}_{b,m}$ defined for $b \geq 2,$ $m\geq 1,$ $b,m \in \mathbb N$, as the fixed point starting in $0$ of the morphism $$\varphi_{b,m}(k)=k(k+1)\dots(k+b-1),$$ where $k \in \{0,1,\dots, m-1\}$ and where the letters are expressed modulo $m$. We use the result of Frid [4] and the study of generalized Thue-Morse words by Starosta [6].

# Keywords:

combinatorics on words, generalized Thue-Morse word, factor frequency

68R15

# References:

1. J.-P. Allouche and J. Shallit: Sums of digits, overlaps, and palindromes. Discrete Math. Theoret. Comput. Sci. 4 (2000), 1-10.   CrossRef
2. L. Balková: Factor frequencies in languages invariant under symmetries preserving factor frequencies. Integers - Electronic Journal of Combinatorial Number Theory 12 (2012), A36.   CrossRef
3. M. Dekking: On the Thue-Morse measure. Acta Univ. Carolin. Math. Phys. 33 (1992), 35-40.   CrossRef
4. A. Frid: On the frequency of factors in a D0L word. J. Automata, Languages and Combinatorics 3 (1998), 29-41.   CrossRef
5. M. Queffélec: Substitution dynamical systems - Spectral analysis. Lecture Notes in Math. 1294 (1987).   CrossRef
6. Š. Starosta: Generalized Thue-Morse words and palindromic richness. Kybernetika 48 (2012), 3, 361-370.   CrossRef