What evidence do we have that the standard analytic tetration $F$ has a natural asymptotic rate of growth even at noninteger arguments?

Intuitively, $y=x$ has a natural growth rate, while $y=x+\sin(x)$ has an oscillating growth rate (despite being monotonic); $y=2^x$ has a natural growth rate, while $2^{x + \sqrt{x} \sin(\log(x))}$ oscillates around $2^x$. We would like to know either that (a) $F$ has a natural growth rate, or (b) some asymptotically incomparable function has a natural growth rate, or (c) the intuitive separation into natural and oscillating growth rates (even when partial and approximate) does not work for functions that resemble tetration or fractional exponentiation.

Recall that $F$ is the unique analytic function defined everywhere on $ℂ$ except on the ray from -2 to -∞ such that $F(0)=1$, $F(\overline z) = \overline{F(z)}$ and $F(z+1)=e^{F(z)}$, and $\lim_{y→∞} F(x+iy) = L_e$, where $L_e≈0.318+1.337i$ is the unique fix-point of the logarithm in the upper half-plane.

The recurrence $F(n+1)=e^{F(n)}$ appears natural, but it does not say whether the relation between (for example) $F(n)$ and $F(n+\frac{1}{2})$ is natural as well, which is important in at least two applications.

Let us define fractional exponentiation, $\mathrm{exp}^a(x) = F(F^{-1}(x)+a)$.
$\mathrm{exp}^a$ for $0<a<1$ fills a gap between (essentially) quasipolynomial and quasiexponential growth rates. However, naturalness of $\mathrm{exp}^a$ depends on the behavior of $F$ at non-integral arguments. For example, there even are smooth $f$ with $f(f(x))=e^x$ that oscillate between polynomial and exponential growth rates. What makes $\mathrm{exp}^a$ (for $a=1/2$ or otherwise) superior over the imitation? For example, do we know of natural (mathematical) processes that grow like $\mathrm{exp}^a(x)$ at large $x$?

Part of the evidence could come from the uniqueness of $F$. Is $F$ the unique analytic function defined everywhere on $ℂ$ except on the ray from -2 to -∞ such that $F(0)=1$, $F(\overline z) = \overline{F(z)}$ and $F(z+1)=e^{F(z)}$? If not, what is the connection between $\lim_{y→∞} F(x+iy) = L_e$, and $F$ and $\mathrm{exp}^a$ having a natural (or well-behaved) asymptotic growth rate for real arguments?

Tetration may also support non-atomic partial (or partially defined) probability measures on $ℕ$. Consistently with ZFC, there is no definable total finitely additive non-atomic probability measure on $ℕ$. However, we can treat $ℕ$ as having an unnormalized probability measure that is valid only when it converges, and we would like to maximize that class while retaining naturalness. Given a function $f$ and a predicate or indicator function $S$, we can set the probability of $S$ using $f$, $P_f(S) = \lim_{n→∞} \frac{∑_{m=1}^n f(m) S(m)}{∑_{m=1}^n f(m)}$ when the limit exists. (A caveat is that if $S$ and $T$ are not disjoint, existence of $P_f(S)$ and $P_f(T)$ does not generally imply existence $P_f(S∪T)$, though with the axiom of choice, there is a total finitely additive extension of $P_f$.) Commonly, $f(m)$ is chosen to be constant, but if $∑_{m=1}^n f(m)$ diverges extremely slowly, for example by setting $f(m) = F^{-1}(m+1)-F^{-1}(m)$, we may get a partial nonatomic measure that is defined for almost all $S$ that we are likely to encounter. The question of naturalness of analytic tetration would be why this choice of $f$ should be preferred over incompatible choices. A starting point is that if we set $f_0(n)=1$, $f_1(n)=1/n$, $f_2(n)=1/(n \log n)$, $f_3(n) = 1/(n \log n \log \log n)$, and so on (the behavior at small $n$ is irrelevant), then (if I understand correctly) $P_{f_i}(S)=P_f(S)$ whenever $P_{f_i}(S)$ exists.

I should note that naturalness of growth rates makes sense intuitively, but is difficult to pin down formally. Arguably, it is the multitude of connections that makes a growth rate natural.

For example, here is evidence that exponentiation has a natural asymptotic growth rate:

  • Exponentiation is a symmetry (mapping addition to multiplication).

  • A large class of functions has growth rates comparable (as opposed to incomparable) with exponentiation. This includes all functions first order definable in $(ℝ,+,e^x)$ allowing parameters (and thus all elementary functions that do not use complex numbers / trigonometry). Here, $f$ and $g$ are comparable iff $(∃x ∀y>x \, f(y)≥g(y)) ∨ (∃x ∀y>x \, g(y)≥f(y))$.

  • Algorithms are much more likely to have asymptotic runtimes $O(2^n)$ as opposed to say $O(2^{n + \sqrt{n} \sin(\log(n))})$, with the later (and not $O(2^n)$) treated as an unusual oscillatory behavior.

There is also a possible pessimistic answer that naturalness of rates of growth does not make sense for tetration. An example evidence supporting that answer would be that having comparable rates of growth is the exception rather than the norm (in contrast to elementary functions). For example, with my current knowledge, I cannot rule out that for the standard $F_2(n+1)=2^{F(n)}$ (and $\mathrm{exp}_2^a(x)=F_2(F_2^{-1}(x)+a)$), there are positive $a≠b$ such that there are arbitrarily large $x$ with $\mathrm{exp}^a(x)=\mathrm{exp}_2^b(x)$, without a good explanation about why $F$ is better than $F_2$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.