Locating factors of a characteristic word via the generalized Zeckendorf representation of numbers

Abstract

Let α be an irrational number with 0<α<1. Let a,b be two distinct letters. The characteristic wordfα of α is an infinite word whose nth letter is a (resp., b) if ⌊(n+1)α⌋−⌊nα⌋=0 (resp., 1), n≥1. For a factor w of fα, the location of w is the set of all positions in fα at which w occurs. The locations of all factors of fα have been determined by Chuan and Ho recently. In this paper, we obtain other formulas for the locations of factors of fα, using the generalized Zeckendorf representation of nonnegative integers. These results are equivalent to the known results obtained by Chuan and Ho in the case α=3−52. We compute the longest common prefix of any two suffixes of fα and compute the order number and location index of each factor of fα, given its length and a position in fα at which it begins.