Let,
a) $f:\mathbb{N}\longrightarrow \left\{0,1,2 \right\}$ or a sequence $f(n):= x_n$ where $x_n\in \left\{0,1,2\right\}$ and $g:\mathbb{N}\longrightarrow \left\{0,1,2 \right\}$ or a sequence $g(n):= y_n$ where $y_n\in \left\{0,1,2\right\}$
b) $f:\mathbb{N}\longrightarrow \left\{0,1 \right\}$ or a sequence $f(n):= x_n$ where $x_n\in \left\{0,1\right\}$ and $g:\mathbb{N}\longrightarrow \left\{0,1 \right\}$ or a sequence $g(n):= y_n$ where $y_n\in \left\{0,1\right\}$
Question-1.
- Is it possible to find such non-trivial functions $f(n)$ and $g(n)$, which gives
$\qquad$ $\qquad$ $\displaystyle \lim_{k \to \infty }\dfrac{\displaystyle \sum_{n=1}^{k} \left(m^n \times f(n) \right)}{\displaystyle \sum_{n=1}^{k} \left(m^n \times g(n) \right)}=\infty$ $\qquad$ and $\qquad$ $\displaystyle \lim_{k \to \infty }\dfrac{\displaystyle \sum_{n=1}^{k} \left(m^n \times g(n) \right)}{\displaystyle \sum_{n=1}^{k} \left(m^n \times f(n) \right)}=0$
where, $m\in\mathbb{Z^{+}}, m \geq 3.$
Or without a unique limit point:
$\qquad$ $\quad$ $\displaystyle \lim_{k \to \infty }\text{sup}\dfrac{\displaystyle \sum_{n=1}^{k} \left(m^n \times f(n) \right)}{\displaystyle \sum_{n=1}^{k} \left(m^n \times g(n) \right)}=\infty$ $\qquad$ and $\quad$ $\displaystyle \lim_{k \to \infty }\text{inf}\dfrac{\displaystyle \sum_{n=1}^{k} \left(m^n \times g(n) \right)}{\displaystyle \sum_{n=1}^{k} \left(m^n \times f(n) \right)}=0$
where, $m\in\mathbb{Z^{+}}, m \geq 3.$
For example let, $f(n)$ and $g(n)$ be constant functions, where $g(n)=0$ and $f(n)=1$ or $f(n)=2$ which works.
But, I`m looking for a non-constant or non-trivial functions which provide the conditions of the problem. How can I find such functions?
But, we can choose infinitely many computable functions, for example $f(n)= n\mod3$ $\quad$ and $g(n)=n^2 \mod3 $ which doesn`t work. In fact, it is much easier to find such functions!
Then, I have $2$ more short questions.
Question-2.
- Is it possible to find infinitely many computable and non-computable functions that provide these conditions?
Question-3.
- Is it possible to find such functions $f(n)$ and $g(n)$ which given by a unique mathematical formula?
This question is related to the question I asked before.
But, the new question is different and expressed more detailed here.
Thank you!
Let Z, be the set of functions from $\mathbb N$ to $\{0,1,2\}$ such that they are ultimately zero constant $$g\in Z \Leftrightarrow \exists N,\forall n\ge N, g(n)=0$$
Then you can easily prove that $f$ and $g$ verify the limits if and only if $g\in Z$ and $f\notin Z$.
Hence, as $Z$ contains only recursive functions, $g$ must be recursive. However, $f$ can be recursive or not.