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If I define $T(c, k, r) = c^{c^{\cdot^{\cdot^{\cdot^{c^r}}}}}$ with $(k-1)$ $c$'s in the tower. I want to understand the asymptotic behaviour of towers with different values of $c$.

I am specifically interested in whether say the behaviour of $T(1.5, k, r)$ is similar/different to that of $T(2, k, r)$. Clearly $c=1$ is a degenerate case, but as far as I can tell, asymptotically it should be the only one for $r,k$ large enough.

Any resources would be appreciated.

Robert S
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  • How do you define $T(c,k,r)$ with non-integer $k$? – Paul Sinclair Apr 07 '20 at 02:05
  • It is only defined for k an integer – Robert S Apr 07 '20 at 02:15
  • But your question specifically asks about the behavior when $k = 1.5$ ("$T(2,1.5,r)$") – Paul Sinclair Apr 07 '20 at 02:16
  • Sorry, my mistake, I'll update it now – Robert S Apr 07 '20 at 02:29
  • As a function of $c$ for fixed $k,r, T(c,k,r)$ is strictly increasing. For $k > 1$, that increase is faster than exponential. It passes through $1$ at $c = 1$. I'm not sure why you call that degenerate (if it were $r$ or $k$ you were varying that would be difference, but there is nothing degenerate about it as $c$ changes). I'm not sure what you are looking for here. – Paul Sinclair Apr 07 '20 at 02:40
  • Maybe degenerate is the wrong word to use, I mean that at $c=1$, the tower does not grow. I'm looking for a general way to compare two towers with different values of $c$. e.g. Is there a general way to bound $T(1.5, k, r)$ using a tower with $c=2$ – Robert S Apr 07 '20 at 18:13
  • As I said $T$ is strictly increasing with $c$, so $T(1.5,k,r) < T(2,k,r)$. What more are you wanting in a bound? – Paul Sinclair Apr 07 '20 at 19:24
  • I would like a lower bound if possible – Robert S Apr 08 '20 at 07:50
  • If you want a lower bound, use a lower value of $c$. At this point I am tired of trying to coax useful information out of you. – Paul Sinclair Apr 08 '20 at 11:57
  • I think it is clear from what I have said that I am interested in a lower bound for $T(1.5,k,r)$ in terms of another tower where $c=2$. I would not ask a question seeking the most trivial bound available. – Robert S Apr 09 '20 at 03:38

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