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Source. Theorem 2.3, 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, acting on $H^2_{ber}(D)/X$ with $X=\operatorname{span}\{1,z,z^2\}$ (p. 3). $A(D)$ is the space of analytic functions on the open unit disc $D$, and $\sqrt2D$ is the open disc of radius $\sqrt2$.

Theorem 2.3 (p. 4). Every $\lambda\in\sqrt2D$ is an eigenvalue of $\mathcal T:H^2_{ber}(D)/X\to H^2_{ber}(D)/X$ of infinite multiplicity. In addition, the paper states, for every $\lambda\in\mathbb C\setminus\sqrt2D$ there is an infinite sequence $\{h_m(\lambda,\cdot)\}_{m\ge1}\subset A(D)$ with $\mathcal Th_m(\lambda,\cdot)=\lambda h_m(\lambda,\cdot)$.

The eigenfunctions are explicit (2.3): with $g(\lambda,z)=\sum_{p\ge0}\lambda^pz^{2^p}$ and $f_m(\lambda,z)=g(\lambda,z^m)$, $h_m(\lambda,\cdot)=f_{6m+4}(\lambda,\cdot)-f_{2m+1}(\lambda,\cdot)$, $m\ge1$.

Read depth. Claims checked: the statement was read clause by clause on p. 4; the proof was read through, not checked step by step.

Proof pointer

Page 4. A direct computation gives $\mathcal Tf_m=\lambda f_m+z^{T(m)}$ (2.4), so the inhomogeneous terms cancel in $h_m$ because $T(2m+1)=T(6m+4)=3m+2$; the $H^2_{ber}(D)/X$ norm of $h_m(\lambda,\cdot)$ is finite when $|\lambda|<\sqrt2$.

Dependencies

None.

Bears on

No Erdős problem page cites it. The paper offers it, in Remark 2.6 (p. 4), as showing that $\mathcal T$ has fixed points not tied to cycles or diverging trajectories; the case $\lambda=1$ gives such fixed points.