In studies of the Collatz conjecture, what research has asserted the existence of a $k$-length cycle and drawn conclusions about its smallest element $m$? In particular, about the behavior of $m$ as $k \rightarrow \infty$?
I know that this question tried, this question speculated, and that the existence of $k$-length cycles given $m$ has been studied a lot. For example
Lagarias (1985) used a result by Yoneda that $m > 2^{40}$ and a theorem by Crandall (quoted as Theorem I) to prove that there are no nontrivial cycles of period length less than $k = 275,000$.
I am interested in the other direction, of properties of $m$ given $k$.
EDIT: I am especially interested in lower bounds for $m$, and whether it has been shown that $m \rightarrow \infty$ as $k \rightarrow \infty$ (thanks to @rukhin for the answer concerning an upper bound on $m$).
Many thanks in advance.
Best Answer
Here is a table, based on computation of my earlier answer (see there). I got the N for which the lower bounds $a_\text{min_1cyc}$ are relatively highest by the continued fractions of $\lambda=\log(3)/\log(2)$.
$a_m$ is the very rough "mean" value of all $a_k$ by ${1\over 2^{S/N}-3}$, such that it is a good upper bound for $a_\min$.
A slightly better (=smaller) upper bound occurs, when the $a_1 \cdots a_N$ are assumed to be packed as tight as possible (though all different), so roughly in steps of $3$, and still solving rhs = lhs. I called it $a_\text{min_compact}$. Because the exact computation is time (or memory)-consumptive I've shown this only for $N \le 10000$
The improvement over $a_m$ however is only marginal, so this reduced display may not be critical.
A lower bound, as you have asked, occurs if we assume the $a_k$ being in a so-called "1-cycle" which means also that they have the widest distance of each other and are consecutive iterates $a_{k+1}=(3a_k+1)/2$.
The minimal $a_k$ of any type of cycle cannot be smaller than $a_\text{min_1cyc}$. I was surprised myself when I saw such large values for the large N ,btw.
A plot of that empirical values suggests, that the square-roots of the maximal $am$ (relevant $N$ indicated by the convergents of the continued fraction of $\log_2(3)$) are roughly equivalent to $N$ (the $a_{min:1cyc}$ even to seem to be equivalent to $N$). (The term "Perigee" comes from Belaga's work which has kindly been linked to by @rukhin)