---

Source. Lemma 2.4, p. 4, of Mikhail Neklyudov, Functional analysis approach to the Collatz conjecture, arXiv:2106.11859v9 (2022), published in Results Math. 79 (2024), no. 4, Paper No. 140, in the edition identified on the source card.

Statement

$T$ is the reduced Collatz map on $\mathbb Z$ and $\mathcal T$ the operator with $\mathcal T(z^n)=z^{T(n)}$, as on the Lemma 1.1 page. With $g(\lambda,z)=\sum_{p\ge0}\lambda^pz^{2^p}$ and $f_m(\lambda,z)=g(\lambda,z^m)$ as in (2.3) (p. 4), so that $f_m(1,z)=\sum_{p\ge0}z^{m2^p}$:

Lemma 2.4 (p. 4). For every diverging sequence $\{T^k(m)\}_{k\ge1}$ of $T$,

$$f_m(1,z)+\sum_{k=1}^\infty z^{T^k(m)}$$

is a fixed point of $\mathcal T$.

Remark 2.5 (p. 4) adds that truncating the trajectory at any index $n$ gives another such fixed point, so a diverging trajectory gives an infinite family.

Read depth. Claims checked: the statement was read clause by clause on p. 4.

Proof pointer

Page 4: it follows from the identity $\mathcal Tf_m=\lambda f_m+z^{T(m)}$ (2.4) at $\lambda=1$; the extra term $z^{T(m)}$ is absorbed by the shift along the trajectory.

Dependencies

Formula (2.4) of the same paper, proved with Theorem 2.3.

Bears on

• #1135: a diverging trajectory of the problem's map would give a fixed point of this form; the lemma does not decide whether one exists.