Logical strength of the linear-operator necessary condition for the Collatz conjecture (Soileau's Fourier-average condition)

Soileau (arXiv:2310.06090) proved that the Collatz conjecture implies a pointwise convergence property of Cesàro means of Fourier series generated by iterating a linear operator $L_1$ that replaces each frequency $n$ by the Collatz value $C(n)$: if every Collatz orbit reaches 1, then for every $|c|<1$ and every angle the means converge to an explicit limit involving the cycle $1\to4\to2\to1$. The paper establishes only necessity. The open problem is to determine the exact strength of this condition: whether the converse of the proved implication holds, i.e. whether convergence of the Cesàro means to the stated limit (for every $|c|<1$ and every angle) already forces every Collatz orbit to reach 1 — equivalently, whether a failure of the Collatz conjecture would necessarily make the convergence condition fail.

Background

The Collatz map $C$ sends a positive integer $n$ to $n/2$ when $n$ is even and to $3n+1$ when $n$ is odd. The Collatz conjecture asserts that iterating $C$ from any positive integer eventually reaches 1, after which the orbit cycles $1\to4\to2\to1$. The conjecture has been verified by computer for all starting values below $2^{68}$ (Barina, 2020), and the survey of Lagarias (1985) remains the standard reference; Tao (2019) later proved that almost all orbits attain almost bounded values, the strongest known partial result. A separate tradition (Berg and Meinardus, 1994) encodes Collatz dynamics in functional equations for generating functions. Soileau's preprint (arXiv:2310.06090) contributes an orbit-encoding by linear-operator iteration, defined as follows. A trigonometric series $f(t)=\sum_{n\ge0}a_n e^{int}$ with $\sum|a_n|<\infty$ (an absolutely convergent Fourier series) carries a linear operator $L_1$ given by $(L_1 f)(t)=\sum_{n\ge0}a_n e^{iC(n)t}$: each frequency $n$ is replaced by the Collatz value $C(n)$, so iterating $L_1$ composes the Collatz map inside the exponents. For a complex parameter $c$ with $|c|<1$, start from the geometric series $f_0(t)=1/(1-ce^{it})=\sum_{n\ge0}c^n e^{int}$; the $n$-th iterate then satisfies $f_n(\theta)=\sum_{r\ge0}c^r e^{iC^n(r)\theta}$, where $C^n$ denotes $n$-fold composition and $C^0(r)=r$. Let $M_n(\theta)=\frac{1}{n}\sum_{j=1}^{n}f_j(\theta)$ denote the Cesàro mean, i.e. the ordinary arithmetic average of the first $n$ iterates. Writing $g_{n,m}(\theta)=\frac{1}{n}\sum_{j=1}^{n}e^{iC^j(m)\theta}$ for the averaged exponential along the orbit of starting value $m$, one has $M_n(\theta)=1+\sum_{m\ge1}c^m g_{n,m}(\theta)$, an absolutely convergent series since $|g_{n,m}|\le1$ and $|c|<1$. Soileau's Theorem 5.2 states: if every Collatz orbit reaches 1, then for every $\theta\in[0,2\pi)$ and every $|c|<1$, $\lim_{n\to\infty}M_n(\theta)=1+\frac{c}{3(1-c)}\left(e^{i\theta}+e^{2i\theta}+e^{4i\theta}\right).$ The proof is elementary: each orbit that reaches 1 eventually repeats with period 3 in the cycle $4\to2\to1\to4$, an averaging lemma fixes the limit of each $g_{n,m}$, and the uniform bound $|g_{n,m}|\le1$ permits interchanging the limit with the sum over $m$. The implication is one-way, so the convergence statement is a necessary condition for the conjecture. Two structural facts frame its strength: at $\theta=0$ every exponential equals 1 and the condition is vacuous, so its content lives at $\theta\neq0$, where it constrains how the exponentials $e^{iC^j(m)\theta}$ distribute along orbits; and the condition concerns a weighted ensemble over all starting values, so a finite set of non-convergent orbits perturbs $M_n(\theta)$ through finitely many weighted orbit averages, making the condition not tautologically equivalent to the conjecture. Nothing is known in the reverse direction: the preprint states no open questions and does not address sufficiency, sharpness, or failure modes.

Problem Statement

Determine the exact logical strength of the necessary condition above, at its natural strength and without added restrictions, so as to decide whether it is equivalent to the Collatz conjecture or strictly weaker than it; since Theorem 5.2 already proves the implication from the conjecture to the condition, incomparability is ruled out and the classification hinges entirely on the converse. Concretely, the open question is the converse of Soileau's Theorem 5.2, taken at the full quantifier strength of that theorem's conclusion: if for every $|c|<1$ and every $\theta\in[0,2\pi)$ the Cesàro means $M_n(\theta)$ converge to $1+\frac{c}{3(1-c)}(e^{i\theta}+e^{2i\theta}+e^{4i\theta})$, must every Collatz orbit reach 1? By contraposition this is the same proposition as the failure question: if the Collatz conjecture is false — through a divergent orbit or a cycle other than $1\to4\to2\to1$ — must the convergence condition fail for some choice of $c$ and $\theta$, or can weighted cancellation across exceptional starting values keep $M_n(\theta)$ at the stated limit?

References

1. Soileau, Kerry M. (2023; revised 16 Aug 2025). 'A Necessary Condition on the Collatz Conjecture'. arXiv:2310.06090 [math.GM], DOI 10.48550/arXiv.2310.06090, https://arxiv.org/abs/2310.06090
2. Lagarias, Jeffrey C. (1985). 'The 3x+1 Problem and Its Generalizations'. The American Mathematical Monthly 92(1), 3-23. DOI 10.1080/00029890.1985.11971528, https://doi.org/10.1080/00029890.1985.11971528
3. Tao, Terence (2019; published 2022). 'Almost all orbits of the Collatz map attain almost bounded values'. arXiv:1909.03562; Forum of Mathematics, Pi 10 (2022), paper e12. DOI 10.48550/arXiv.1909.03562; journal DOI 10.1017/fmp.2022.8, https://arxiv.org/abs/1909.03562, https://doi.org/10.1017/fmp.2022.8
4. Barina, David (2020, online; issue 2021). 'Convergence verification of the Collatz problem'. The Journal of Supercomputing 77(3), 2681-2688. DOI 10.1007/s11227-020-03368-x, https://doi.org/10.1007/s11227-020-03368-x
5. Berg, Lothar; Meinardus, Günter (1994). 'Functional Equations Connected With The Collatz Problem'. Results in Mathematics 25(1-2), 1-12. DOI 10.1007/bf03323136, https://doi.org/10.1007/bf03323136