A binary word is called $q$-decreasing, for $q>0$, if inside this word each of length-maximal (in the local sense) occurrences of a factor of the form $0^a1^b$, $a>0$, satisfies $q \cdot a > b$. We bijectively link $q$-decreasing words with certain prefixes of the cutting sequence of the line $y=qx$. We show that for any real positive $q$ the number of $q$-decreasing words of length $n$ grows as $C_q \cdot \Phi(q)^n$ for some constant $C_q$ which depends on $q$ but not on $n$. From previous works, it is already known that $\Phi(1)$ is the golden ratio, $\Phi(2)$ is equal to the tribonacci constant, $\Phi(k)$ is $(k+1)$-bonacci constant. We prove that the function $\Phi(q)$ is strictly increasing, discontinuous at every positive rational point, and exhibits a fractal structure related to the Stern-Brocot tree and Minkowski's question mark function.
@article{10_37236_12705,
author = {Sergey Dovgal and Sergey Kirgizov},
title = {Structure and growth of {\(\mathbb{R}\)-Bonacci} words},
journal = {The electronic journal of combinatorics},
year = {2025},
volume = {32},
number = {3},
doi = {10.37236/12705},
zbl = {8097660},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12705/}
}
TY - JOUR
AU - Sergey Dovgal
AU - Sergey Kirgizov
TI - Structure and growth of \(\mathbb{R}\)-Bonacci words
JO - The electronic journal of combinatorics
PY - 2025
VL - 32
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/12705/
DO - 10.37236/12705
ID - 10_37236_12705
ER -
%0 Journal Article
%A Sergey Dovgal
%A Sergey Kirgizov
%T Structure and growth of \(\mathbb{R}\)-Bonacci words
%J The electronic journal of combinatorics
%D 2025
%V 32
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/12705/
%R 10.37236/12705
%F 10_37236_12705
Sergey Dovgal; Sergey Kirgizov. Structure and growth of \(\mathbb{R}\)-Bonacci words. The electronic journal of combinatorics, Tome 32 (2025) no. 3. doi: 10.37236/12705
