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.

Discrete connections (part XI)

This part can be better classified as a "bigger picture" overview. We've seen in the past articles several strong hints concerning the global properties of the euclidean summation but no word yet about local ones. Here we try to give some of these local properties a go, while raising subtle bridges between the euclidean summation and the euclidean fractal.

As accustomed, we will define some derivatives of the euclidean summation peak and euclidean summation positional peak:

[; \tau_2^{p}(n) = \left\{ \begin{array}{l} 0, \; if \; I_n^{p} = \emptyset \\ \max\{ \tau_1(n,k) \; | \; k \in I_{n}^{p} \}, \; otherwise \end{array} \right. ;]

[; \tau_3^{p}(n) = \left\{ \begin{array}{l} 0, \; if \; I_{n}^{p} = \emptyset \\ \max\{ k \in I_{n}^{p} \; | \; \tau_1(n,k) = \tau_2^{p}(n) \}, \; otherwise \end{array} \right. ;]

where:

[; p \in \mathbf{N}, \; p \ge 1 ;]

[; I_{n}^{p} = \{k \in \mathbf{N} \; | \; \frac{n}{p+1} \le k \le \frac{n}{p} \} ;]

The target conjectures for these derivatives are:

[; \lim_{n \rightarrow \infty} \frac{\tau_2^{p}(n)}{n} = \frac{\varphi}{p + \phi} ;]

[; \lim_{n \rightarrow \infty} \frac{\tau_3^{p}(n)}{n} = \frac{1}{p+\phi} ;]

For p=1 these conjectures translate to the result proven for the euclidean summation peak and respectively to the conjecture for the euclidean summation positional peak.

We can go even deeper with this construction, and involve yet another classic: continued fractions (and thus tighten the links with the euclidean fractal). Indeed:

[; \forall s \in \mathbf{N}, \; s > 0 \; and \; \forall w = (p_1, \ldots, p_s) \in (\mathbf{N} \setminus \{0\})^{s} ;]

let us define:

[; \tau_2^{w}(n) = \left\{ \begin{array}{l} 0, \; if \; I_n^{w} = \emptyset \\ \max\{ \tau_1(n,k) \; | \; k \in I_{n}^{w} \}, \; otherwise \end{array} \right. ;]
[; \tau_3^{w}(n) = \left\{ \begin{array}{l} 0, \; if \; I_{n}^{w} = \emptyset \\ \max\{ k \in I_{n}^{w} \; | \; \tau_1(n,k) = \tau_2^{w}(n) \}, \; otherwise \end{array} \right. ;]

where

[; I_{n}^{w} = \{k \in \mathbf{N} \; | \; \cfrac{n}{p_1+\cfrac{1}{\ddots + \cfrac{1}{p_s + \epsilon(s)}}}} \le k \le \cfrac{n}{p_1+\cfrac{1}{\ddots + \cfrac{1}{p_s + 1 - \epsilon(s)}}}} \} ;]

and:

[;\epsilon(s) = \left\{ \begin{array}{l} 0, \; if \; s \; even \\ 1, \; otherwise \end{array} \right. ;]

The conjectured results for these generalized derivatives are:

[; \lim_{n \rightarrow \infty} \frac{\tau_3^{w}(n)}{n} = \cfrac{1}{p_1+\cfrac{1}{\ddots + \cfrac{1}{p_s + \phi }}} ;]

Next: To be continued ...