03/28/2018, 07:57 PM
(This post was last modified: 03/28/2018, 09:33 PM by sheldonison.)

Dmytro Taranovsky asks in his post, https://mathoverflow.net/questions/28381...rent-bases ... "

... Let a and b be real numbers above e^{1/e}, and c<d be real numbers. Do we have

\( \lim_{x \to \infty}\;\frac{\exp_a^c(x)} {\exp_b^d(x)} \;= 0 \)

... The relation with this question is that if the above limit holds, it gives evidence that fractional exponentiation provides a natural growth rate intermediate between (essentially) quasipolynomial and quasiexponential.

This question seems to have lots of room for discussion so I thought I would post it here. Here are my conjectures related to this question.

(1) If you use Peter Walker's http://eretrandre.org/rb/files/Walker1991_111.pdf Tetration solution appropriately extended to all real bases > exp(1/e), then my conjecture is that the limit holds. Unfortunately, Peter Walker's solution is also conjectured to be nowhere analytic; see: https://math.stackexchange.com/questions...lytic-slog

(2) That for Kneser's solution, there are counter examples and the limit does not hold, even with the restriction that c<d, even with the additional restrictions that a<b, there are cases where x is arbitrarily large and

\( \exp_a^c(x)\;>\;\exp_b^d(x) \)

(3) Given any analytic solution for base(a), then if we desire a tetration solution for base(b) which has the desired property then I conjecture that the tetration base b is nowhere analytic! The conjecture is also that the slog for b would be given by a modification of Peter Walker's h function.

... Let a and b be real numbers above e^{1/e}, and c<d be real numbers. Do we have

\( \lim_{x \to \infty}\;\frac{\exp_a^c(x)} {\exp_b^d(x)} \;= 0 \)

... The relation with this question is that if the above limit holds, it gives evidence that fractional exponentiation provides a natural growth rate intermediate between (essentially) quasipolynomial and quasiexponential.

This question seems to have lots of room for discussion so I thought I would post it here. Here are my conjectures related to this question.

(1) If you use Peter Walker's http://eretrandre.org/rb/files/Walker1991_111.pdf Tetration solution appropriately extended to all real bases > exp(1/e), then my conjecture is that the limit holds. Unfortunately, Peter Walker's solution is also conjectured to be nowhere analytic; see: https://math.stackexchange.com/questions...lytic-slog

(2) That for Kneser's solution, there are counter examples and the limit does not hold, even with the restriction that c<d, even with the additional restrictions that a<b, there are cases where x is arbitrarily large and

\( \exp_a^c(x)\;>\;\exp_b^d(x) \)

(3) Given any analytic solution for base(a), then if we desire a tetration solution for base(b) which has the desired property then I conjecture that the tetration base b is nowhere analytic! The conjecture is also that the slog for b would be given by a modification of Peter Walker's h function.

- Sheldon