Consider a function: $f: \Bbb N \to \Bbb N$ defined as follows:
$$f(n)=\bigg\lbrace\#\lbrace a_t(x) \rbrace >\#\lbrace b_t(x) \rbrace: \# \pitchfork \mathrm{id}\bigg \rbrace.$$
This notation means we're counting the number of curves of the family $a_t(x)$ which are greater than the number of curves of the family $b_t(x)$ up to a given magnitude, and the counting is being done along the identity path. You could specify a different path alternatively but I'd like to stick to this path.
Let $a_t(x) \circ a_t(x)=\mathrm{id}$ and $b_t(x)\circ b_t(x)=\mathrm{id}.$
Define functions:
$$ b_t(x):=\exp\bigg(\frac{\log^2(t)}{\log x}\bigg). $$
and $$ a_t(x):=\frac{t}{x}. $$
Here's an example for when the index set $t=\big\lbrace 1/10,2/10,3/10,4/10,5/10,6/10,7/10,8/10,9/10 \big\rbrace.$ You get the following graph:
It shows that $f(1)=1,f(2)=3,f(3)=5,f(4)=6,f(5)=7,f(6)=8,f(7)=9,f(8)=9,f(9)=9.$
Now you can iterate the density and go one more level with the index set. Now, set $t=\big\lbrace 1/100,2/100,3/100,\cdot\cdot\cdot,99/100 \big\rbrace$ You get this picture:
I want to take this process ad infinitum and understand something. I noticed a pattern. $f(n)$ for $1\le n \le k $ is an odd number. Then after hitting the threshold $k$ there's a string of even numbers.
I'm just wondering about something:
Why does hitting $k$ change the sequence from odds to evens? And what can be said about the function and the pattern as the index set grows very large?