A possibly easy question about latent geometry in Collatz sequences

Question

I have a question about some (seemingly unimportant) behavior I noticed in Collatz sequences, which I haven't been able to find a general answer to upon rough scan of the literature (please be aware that this is not my field, so there may be some ignorance here).

A length-$m$ Collatz tuple is an $m$-tuple of the form $(c_0, c_1, \dotsc, c_{m - 1})$, where $c_0 \in \mathbb N$ and $c_{i + 1}$ is the Collatz iterate of $c_i$ for all $i$.

It appears that all length $m$ Collatz tuples lie on one of $m$ distinct lines in $m$-dimensional space, where the angle between each pair of such lines is $\pi/4$. This fact is clear for $m=2$, where all length-2 Collatz tuples lie on the lines $y = 3x+1$ and $y = x/2$ which have an intersection angle of $\pi/4$.

That this necessarily holds for the general $m$ case however is not immediately obvious to me (where we take Collatz orbits of length greater than or equal to $m$). I have been able to come up with a variety of arguments for why this may be the case, but none have been entirely convincing or neat. Can someone elucidate why this appears to be so, and if this property fails for some $m$, why? Can you provide sketch of proof (unless I am missing something basic)?

Answer 1

I see the following arithmetical background of your conjecture.

Put $\Lambda=\{1/2,3\}^m$. For each $\lambda=(\lambda_0,\dots,\lambda_{m-1})\in\Lambda$ we define a vector $r'_\lambda=(r_{\lambda,0},\dots, r_{\lambda,m-1})$ as follows. Put $I(\lambda)=\{0\le i\le m-1:\lambda_i=3\}$. If $I(\lambda)$ is empty then put $r'(\lambda)=0$, otherwise put $i=\min I(\lambda)$, $r'_{\lambda,k}=0$ for $k<i$, $r'_{\lambda,i}=1$, and for each $i+1\le k\le m-1$ let $r'_{\lambda,k}$ equals to $r'_{\lambda,k}/2$, if $\lambda_k=1/2$, and equals to $3r'_{\lambda,k}+1$, if $\lambda_k=3$.

It is easy to see that for each Collatz $m$-tuple $c=(c_0,\dots,c_{m-1})$ there exists $\lambda\in\Lambda$ such that $c=\lambda c_0+ r’_\lambda$. That is $c$ belongs to a line $r’_\lambda+\lambda t$, $t\in\Bbb R$. It can be well-known for which $\lambda\in\Lambda$ there exists a Collatz $m$-tuple $c=(c_0,\dots,c_{m-1})$ such that $c=\lambda c_0+ r’_\lambda$. The set of all such $\lambda$ provides a family $L$ of lines, containing all Collatz $m$-tuples. Moreover, if a line $\ell’$ in $\Bbb R^m$ contains at least $2^m+1$ Collatz $m$-tuples then $\ell’$ has two common points with some line $\ell\in L$, so $\ell’=\ell$.