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I am currently reading the work of Belaga on upper bounds on minimal cyclic iterates in the $3x+d$ problem.

In the paper, the author gives an upper bound on the perigee as $$ dk^{C_2} $$ where $k$ is the number of odd elements in the cycle, and $C_2$ is an effectively computable constant not exceeding $32$ (as per Corollary 2 in the cited paper of Laurent et al).

Later in the introduction, Belaga mentions an upper bound for the apogee (maximal element)

$$dk^C (3/2)^k $$

(the author and Mignotte derive this upper bound in another paper )

Question: Does 32 still apply as an upper bound for the effectively computable constant C (when bounding the apogee)?

The author writes that he corresponded with another author in the derivation of this constant (for the perigee).

rukhin
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  • Hmm, empirically, $C_2$ seems to be around $2$ from own heuristical computations (a short list is in my entry https://math.stackexchange.com/a/2886523/1714 ), but that -although much suggestive- does of course not provide a proof... – Gottfried Helms Jan 29 '19 at 10:56

1 Answers1

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One can apply the results of Rhin (as provided by Lemma 12 in the work of Simons and De Weger) to derive sharp constants.

Assume $k+l>k$. Lemma 12 in Simons/De Weger demonstrates the inequality

$$ (k+l)\log 2 - k\log 3 > e^{-13.3(0.46507)}k^{-13.3}.$$ This inequality provides means for deriving a lower bound on the denominator $2^{k+l}-3^{k}$ of a periodic orbit element; the argument in the abovementioned paper of Belaga/Mignotte demonstrates how this lower bound can be applied to derive an upper bound on the maximal iterate element.

rukhin
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  • One problem I still have with this nice formula is that I didn't see a route, *how* these constant are derived (at least as a general idea, understandable by an amateur) . [Of course it is nice to have such nice constants already at hands :-) ( +1 for the application here)] – Gottfried Helms Apr 22 '18 at 10:28