By G. Rhin, cited by John Simons, 2007, we have the upper bound for
$$ |S \log2 – N \log 3 | \gt \exp(-13.3(0.46057+\log(N))) \qquad \text{roughly:} {1\over 457 N^{13.3}}
$$
This has been used by John Simons to disprove the 1-cycle in the Collatz ($3x+1$)-Problem.
I'm fiddling with the equivalent question in the $5x+1$ – problem. The 1-cycle here has already been handled by R. Steiner in 1981, and he disproved the existence of any 1-cycle for odd-step-length $N>3$ (the 1-cycles with $N=2$ and $N=3$ are well known), but it is very complicated for me to read the part with the A.Baker-based bounds, and I would like to apply instead a G. Rhin-like estimate for the lower bounds of
$$ |S \log2 – N \log 5 | \gt ??? $$
I'm until now unable to apply and/or modify the underlying results of A. Baker myself accordingly.
So my questions:
- Can I use (at least for large $N$) the given bound analoguously?
- Or what would be an adapted bound?
If I could use that bound, it would be possible to disprove the 1-cycle for the $5x+1$-problem much elementary with the need of direct checks only for $N=4 \ldots 104 $ (1-cycles with $N=2$ and $N=3$ exist and are well known)
If details of my approach (and thus for my needs) are wished, see also my "1-cycle for the $3x+1$" – text at my homepage
Simons, John L., On the (non-)existence of (m)-cycles for generalized Syracuse sequences, Acta Arith. 131, No. 3, 217-254 (2008). ZBL1137.11016.
In Simons' article cited: Rhin, Georges, Approximants de Padé et mesures effectives d’irrationalité. (Padé approximants and effective measures of irrationality), Théorie des nombres, Sémin. Paris 1985/86, Prog. Math. 71, 155-164 (1987). ZBL0632.10034.
Steiner's disproof of the 1-cycle in the $5x+1$-problem: Steiner, Ray, On the ”QX+1 problem,” Q odd, Fibonacci Q. 19, 285-288 (1981). ZBL0474.10005.
Best Answer
I just found the remark of J.Simons (2007) where he refers to an estimate of M. Laurent, M. Mignotte and Yu. Nesterenko (1995), which is weaker than that of G. Rhin, but is usable for the $px+q$-case.
This is cited/used in
[Si2007] John L Simons, 2007 On the (non)-existence of m-cycles for generalized Syracuse sequences 22nd November 2007, online version
Eq (38) leads to the formula for the 1-cycle in the $5x+1$-problem (translated to my notation scheme):
$$ (\Lambda =) \qquad |S \log 2 - N \log 5|> \exp(−24.34·\log 5·(\max(\log( N)+1.057,21))^2) $$ (I think, I can improve this formula for the cases where $\log (N) + 1.057<21$ by simple empirical observations, perhaps I'll insert that later here)
This solves the question and also gives a tool for improvement
(If someone shall come up with a better estimate, then of course I'd appreciate and "accept" that follow-up answer)
snippet from screenshot of pg 16/17.
Note: The Simons' notation ($K,K+L,p,q$) is my notation $N,S,5,1$