Discrete connections (part XII)

First a small stop to explain this sudden "gold rush" about finite continued fractions and how they impact upon the subject at stake.

1. As mentioned previously, they are related to the euclidean fractal by its construction. Indeed, consider:

[; the \; (p_{s}-1)^{th} \; block \; of \; \ldots \; of \; the \; (p_{1}-1)^{th} \; block \; of ;]
[; the \; euclidean \; fractal \; defined \; by \; u=0, v=i, w=1+i, z=1 ;]

then its base endpoints' real parts are:

[;\cfrac{1}{p_1+\cfrac{1}{\ddots + \cfrac{1}{p_s}}}} \; and \; \cfrac{1}{p_1+\cfrac{1}{\ddots + \cfrac{1}{p_s + 1}}}};]

2. As you might already know, any positive rational number can be expressed as a continued fraction whose components are, in a top-down representation, the successive quotients obtained through Euclid's gcd algorithm applied to the numerator and denominator. Moreover, there is only one such canonical representation, i.e. if a sub-unitary rational number k/n is equal to a canonical finite continued fraction then we have all the steps of Euclid's gcd algorithm (we have all the quotients) for n and k.

Further more, if we strictly bound a positive sub-unitary rational number k/n to values such as described in 1, then Euclid's gcd algorithm for n and k has at least s steps, for which the quotients are given by the continued fractions defining the boundaries.