A Run-hierarchical Description of Upper Mechanical Words with Irrational Slopes Using Continued Fractions
Hanna Uscka-Wehlou · KTH Publication Database DiVA (KTH Royal Institute of Technology) · 2008
The main result is a run-hierarchical description (by continued fractions) of upper mechanical words with positive irrational slope less than 1 and intercept 0. We compare this description with two classical methods of forming of such words. In order to be able to perform the comparison, we present a quantitative analysis of our method. We use the denominator of the convergents of the continued fraction expansion of the slope to compute the length of the prefixes obtained by our method. Due to the special treatment which is given to the elements equal to 1, our method gives in some cases longer prefixes than the two other methods. Our method reflects the hierarchy of runs, by analogy to digital lines, which can give a new understanding of the construction of upper mechanical words.