A version of the Shooting Room "paradox" involves selecting disjoint sets in the plane, first selecting a set of area $1$, then a set of area $r$, then of $r^2$, etc.--i.e. geometric growth--eventually stopping after selecting, say, the $t^\text{th}$ set of area $r^{t-1}$. The proportion of area in the final set, relative to the total area selected, is then given by the following ratio: $$F_r(t) = { r^{t-1} \over 1 + r + r^2 + \cdots + r^{t-1}}={1-r^{-1} \over 1-r^{-t}} \tag{1}$$ for which it is apparent that $F_r(t) > 1-r^{-1}$ for any positive real $r$ and any positive integer $t.$ E.g., if $r=10,$ then the final set constitutes more than $90\%$ of the total selected area, regardless of the value of $t$ or how it is chosen.
Now suppose we generalize (1) to hyperoperations, letting $[k]$ denote the $k$th operation in the sequence $(+,\times,\uparrow,\uparrow\uparrow,\ldots)$: $$F^{(k)}_r(t) ::= { r[k](t-1)\over r[k]0 + r[k]1 + r[k]2 + \cdots + r[k](t-1)}\tag{2},$$ where again $t$ is any positive integer, but now $r$ is allowed to be any positive real only when $k\le 4$, otherwise $r$ is restricted to the positive integers.
Thus $k=3$ gives the original (1), and the remainder of this post concerns only the case $k=4$ (tetration), for which
$$F^{(4)}_r(t) ::= { ^{t-1}r\over ^0r +\,^1r +\,^2r + \cdots +\,^{t-1}r},\tag{3}$$
where $^tr$ denotes an exponential tower of height $t$.
This case turns out to have perhaps a surprising amount of structure -- here's a picture showing $F^{(4)}_r(t)$ for a range of $20$ various $r$-values with $t\in\{1,...,15\}$:
In this picture, the line-plots have $r$-values that generally decrease from top to the bottom (e.g. at $t=2$), as follows:
Blue[8, 5, 3, 2, 1.6, 1.5, 1.45] Magenta[1.4, 1.3, 1.2, 1.1, 1] Red[0.9, 0.7, 0.5, 0.35, 0.2, 0.1, 0.06, 0.01]
Distinct behaviors are clearly discernible in three separate regions:
$\color{blue}{\text{Blue}}$: A "bowl" shape with initial decrease eventually followed by monotonic convergence to $1.$ Region(?): $r_{\text{crit}}<r<\infty,$ where $r_{\text{crit}}=e^{1/e}=1.44466786...$
$\color{magenta}{\text{Magenta}}$: Monotonic decrease to $0.$ Region(?): $1\le r \le r_{\text{crit}}.$
$\color{red}{\text{Red}}$: A "sawtooth" shape converging to $0.$ Region(?): $0<r<1.$
Question 0: Are my computations correct? I.e., would someone please confirm that these behaviors are real and not due to some numerical strangeness?
Question 1: What are the exact $r$-intervals defining these three behavioral regions (conjectured above)? (EDIT: See my footnote at the end.)
NB-1: I wrote the above description to agree with the color choices in my picture, but I think some got mis-colored, because three of the Red cases look monotonic. It appears that the boundary between monotonic decreasing and "sawtooth" should be somewhat less than $r=1$, but I haven't figured out what it is. Interestingly, according to Wikipedia the interval of convergence for $^\infty r$ is $[e^{-e},e^{1/e}]$, but (unlike the upper end-point, which is my $r_\text{crit}$) the lower end-point doesn't seem to play a role in distinguishing the three behavioral regions (but I may be mistaken).
NB-2: It's a (bad) play on words to note that hyperoperations have here resulted in hyperbolic functions. The dotted line-plot in the picture is for the case $r=1$, which is exactly the hyperbola ${1\over t}.$ Also, although I didn't include this in the picture, the sawtooth shapes appear to be bounded above by the hyperbola $2\over 1+t$ (the upper teeth just touching it), and seem to become centered on the hyperbola $1\over 1+t$ as $t\to\infty.$
Question 2: Can a formula be found for the location $t_\text{min}$ of the "bottom of the bowl", i.e, the minimum, given $r\in (r_{\text{crit}},\infty)$?
Here are some computed minima as $r$ approaches the critical value (for reference, I include t_half, the value of $t$ when $F^{(4)}_r(t)$ first increases to at least ${1\over 2}$):
r t_min t_half
-------------- ------ ------
2. 3 4
1.5 10 13
1.45 36 44
1.445 159 177
1.4447 538 571
1.44467 2148 2214
1.444668 8556 8688
1.4446679 16401 16403
1.44466787 34001 34160
1.444667862 102911 102928
1.4446678611 340951 340971
1.4446678610... = r_crit
The listed $r$-values approach $r_\text{crit}$ from above (so minima exist), and are such that decreasing the rightmost digit would produce a value less than $r_\text{crit}$ (so the minimum would not exist, as the function then decreases monotonically to $0$ instead of increasing toward $1$).
EDIT:
Footnote: The cited Wikipedia article says that $[c_0,c_1]= [e^{-e},e^{1/e}]$ is the interval of convergence for $^\infty r$, which divides the range of $r$ into the following three regions, and I've added a description consistent with what I've observed numerically for each:
- $r\in(c_1,\infty)$: In this region $^tr$ increases monotonically without bound. In this case, we can show that $\lim_{t\to\infty}F^{(4)}_r(t)=1$ (thus explaining the "bowl" shapes) by proving that ${^0r +\,^1r +\,^2r + \cdots +\,^{t-2}r\over ^{t-1}r}\to 0.$
- $r\in[c_0,c_1]$: In this region $^tr$ converges to a point, say $x_r$, but does not do so monotonically; rather, the $^tr$ sequence alternately increases and decreases from one $t$-value to the next. (As illustrated in the WP article, $x_r\in[x_{c_0},x_{c_1}]=[1/e, e]$.)
- $r\in(0,c_0)$: In this region $^tr$ does not converge, but the two disjoint sequences $({^0}r,{^2}r,{^4}r,...)$ and $({^1}r,{^3}r,{^5}r,...)$ do converge (monotonically, to two different limits, say $a_r$ and $b_r$, respectively). Thus the $^tr$ sequence alternates between these two convergent subsequences. (It appears that as $r\to 0$, $a_r\sim 1-r\to 1$ and $b_r\sim r\to 0$.)
I think the behavior of $^tr$ in the last two of these regions accounts for the monotonic and the "sawtooth" behavior seen in the plots of $F^{(4)}_r(t)$, but it isn't yet clear how to prove it.