Consider the set $i_0,i_1,...$ defined as
$i_0$ is the smallest element and $i_0 = 0$.
If $ (i_a - i_b)^2 < 1$ then $i_c$ is Also in the set and given by
$$ i_c = (i_a + i_b + 1)/2$$ [*]
Let $T(x,y)$ be the cardinality of $i_j$ Numbers in the interval $[x,y]$. Notice $1/2,3/4,7/8,...$ are all in the set hence $T(x,y)$ can be infinite and therefore I can not " simply say " $T(x,y) = T(0,y) - T(0,x) $ Because of the case $ \infty - \infty$.
However the set is not dense like the rationals and we can Find a nonzero gap between any pair of elements in the set.
Apparantly the gap between the real $x$ and the smallest next element $i_k$ is given by
for $ x < 0 : $ $$ f(x) = -x$$
for $x > 0 :$ $$ f(x) = f(x - f(x-1))/2 $$ [ ** ]
It turns out
$f(0) = 2^{-1} $
$f(1) = 2^{-3} $
$f(2) = 2^{-10}$
$f(3) = 2^{- 1541023937}$
And this sequence must Go to zero very fast !
Tommy1729 claimed for large $H$ , $ - ln(f(K)) > exp^{[K]}(1)$ where ^[ .. ] means iteration. Hence going to 0 faster Then tetration ( base e).
( I looked at the tetration forum and Some ackermann type papers but could not Find this )
Now I assume the function $f$ is hard to compute and can not be computed differently from Its definition above.
I assume the function $T(x,y) $ is Also hard to compute. I assume $T(x,y)$ does not help much for computing $f $, Unless perhaps $T(x,y)$ is small.
In fact I think computing $T(x,y)$ can only be done by multiple computations of $f$ 's !!?
And I think we need a way to test for $T(x,y) =$ oo.
Many questions.
But the main 2 questions for here are the following.
[ I have to restrict to 2 , since the answer would be too long ; better asking others in a seperate post ]
How does one show [ * ] implies [ ** ] ? How does one show [ ** ] implies [ * ] ?
Btw What field of math is this ? Does not feel like combinatorics to me ...
References are very Welcome too !!
Thanks in advance.
