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

摘要

Let α be an irrational number with 0 < α < 1. Let a, b be two distinct letters. The characteristic word f_α 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 f_α, the location of w is the set of all positions in f_α at which w occurs. The locations of all factors of f_α have been determined by Chuan and Ho recently. In this paper, we obtain other formulas for the locations of factors of f_α, 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 = 3-5~(1/2)/2. We compute the longest common prefix of any two suffixes of f_α and compute the order number and location index of each factor of f_α, given its length and a position in f_α at which it begins.