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 ...