How fast does the Syracuse characteristic function decay? Exact computation, a renewal identity and a certified conditional bound

1. Problem and contribution

Tao's theorem that almost all Collatz orbits attain almost bounded values [1] rests on one Fourier estimate. Let $a_1,a_2,\ldots$ be i.i.d. geometric random variables with $\mathbb{P}(a=k)=2^{-k}$ for $k\geq 1$, let $s_m=a_1+\cdots+a_m$, and define the Syracuse random variable

$$ \mathrm{Syrac}_n \;=\; \sum_{m=1}^{n} 3^{m-1}\,2^{-s_m} \pmod{3^n}, $$

with characteristic function $\chi_n(\xi)=\mathbb{E}\, e\!\left(\xi\,\mathrm{Syrac}_n/3^n\right)$, where $e(x)=e^{2\pi i x}$. (Tao indexes the digits in the opposite order; because the $a_i$ are i.i.d., the two laws coincide.) Tao proves $|\chi_n(\xi)|\ll_A n^{-A}$ for every $A$, uniformly in $\xi$ coprime to $3$ [1, Prop. 1.17]. By a remark in [1], this estimate is equivalent to his fine-scale mixing proposition, since $|\chi_n(\xi)|\le\mathrm{Osc}_{n-1,n}$ for $3\nmid\xi$. Tao also remarks that a random-map heuristic predicts mixing at rate $\exp(-cm)$, which he does not attempt to establish. Neither the size of $c$ nor the frequencies and mechanism that set it appear to have been studied.

This paper studies the true size of

$$ M_n \;=\; \max_{3\nmid \xi}\,|\chi_n(\xi)| . $$

We do not claim any improvement of Tao's Collatz theorem. In his argument the Fourier input already costs less than any power of $\log x$, and the density exponent is limited by archimedean steps (§6, Scope). The interest of $M_n$ is intrinsic: it measures how fast the 3-adic Syracuse law mixes, and the answer differs sharply from the random-map prediction.

Since $2$ is a primitive root modulo $3^n$, every unit is a power $2^j$ (with $-1=2^{3^{n-1}}$), and $\chi_n(-\xi)=\overline{\chi_n(\xi)}$. We write extremal frequencies as $\pm2^j$ with $0\le j<3^{n-1}$. The contributions are:

1. Exact values. We compute $\chi_n$ exactly on all $2\cdot 3^{n-1}$ units for $n\le 18$ with an $O(3^n)$ recursion. For individual frequencies we use a new $O(n^2)$ dynamic program, which evaluates $\chi_n(2^ju)$ exactly for $n$ up to $2000$. In float64 the relative error is below $10^{-9}$, as confirmed by a certified recomputation for $\ell\le800$. For $n\le18$ the maximum over all units is attained at $\xi=\pm2^{j}$ with $j=\lambda n-O(1)$, $\lambda=\log_2 3$. Along this family the maximum $M^*_n$ falls from $1.1\times10^{-2}$ ($n=18$) to $2.2\times10^{-52}$ ($n=2000$). Free fits give exponential rate $\kappa=0.0552$–$0.0553$. With $\kappa=I$ fixed, the effective polynomial exponent rises from $1.0$ to $1.35$ as $n$ grows from $200$ to $2000$.
2. A renewal identity (Theorem 1). For all $j\ge0$ and $u$ coprime to $3$, $\chi_n(2^j u)$ equals an explicitly computable "cold-walk" expectation plus a sum $\sum_\ell w_\ell(j,u)\,\chi_\ell(u)$. The term for $\ell$ is weighted by the probability that the partial sums first exceed $j$ at step $n-\ell+1$.
3. A conditional transfer theorem (Theorem 2). If $|\chi_\ell(1)|\le B\rho^\ell$ for some $\rho<\lambda-1\approx 0.585$, then $|\chi_n(2^j)|\le C\,e^{-I n+\theta^*(j-\lambda n)}$, where

$$ I=\lambda\bigl(\ln 2-H(1/\lambda)\bigr)\approx 0.05498,\qquad \theta^*=\ln\frac{\lambda}{2(\lambda-1)}\approx 0.3036, $$

and $H$ is the natural-log binary entropy. The rate $I$ is the Cramér cost for a walk with mean step $2$ to stay below the neutral-growth line $s_m\le\lambda m$. 4. Certified verification of the hypothesis to $\ell=1000$ (Corollary 3). Exact values give $|\chi_\ell(1)|\approx e^{-0.565\ell}$, i.e. $\rho_{\rm eff}=0.568$, inside the threshold $\lambda-1=0.585$. Explicitly, $|\chi_\ell(1)|\le 5.70\cdot 0.58^\ell$ for all $\ell\le1000$. This bound is certified by ball arithmetic (Corollary 3). So for $n\le1000$ the conclusion of Theorem 2 holds rigorously, with explicit constant $C=39$ for all $j$ simultaneously; it is nontrivial for $67\le n\le1000$ and $j\lesssim1.7n$. Certified lower bounds at the extremal $j^*$ show that it is sharp up to a factor of at most $1.6\times10^4$ at $n=100,\dots,1000$. The margin against the threshold is small: $\ln((\lambda-1)/0.568)=0.029$ nats per level. This explains the long pre-asymptotic regime, in which effective exponents fitted at $n\le62$ are misleading. 5. Cross-checks. Exact-integer Monte Carlo ($n\le62$) and a hybrid renewal model (exact weights plus random phases) agree with the exact values to within sampling error and $10\%$ respectively. 6. Conjecture C1: $M_n=e^{-In+O(\log n)}$, so in particular $\frac1n\log M_n\to -I$. Empirically, $n\,e^{In}M^*_n$ stays at $0.32\pm0.01$ for $180\le n\le800$ and then decreases to $0.25$ at $n=2000$. So the polynomial correction is not yet determined. The extremal frequencies are $2^{\lambda n-6.5+O(1)}$. These frequencies correspond to the neutral-growth walks $2^{s_m}\approx3^m$, which also govern heuristic models of Collatz cycles. The rate $I$ is ten times smaller than the rate $\tfrac12\ln 3$ predicted by a random-map model.

We also explain (Section 6) why renewal-type arguments naturally output polynomial decay, as in Tao's theorem: they give $M_n\lesssim c' M_{\approx 0.21n}$.

2. Related work and distinction

Tao [1] introduced $\mathrm{Syrac}_n$ and proved the polynomial bound $|\chi_n(\xi)|\ll_A n^{-A}$. His proof pairs consecutive digits, which produces factors $|\cos|$ that are bounded away from $1$ whenever a phase $\{\xi 3^{m}2^{-s}/3^n\}$ lies away from $0$. He then shows that the set of "black" positions, where the phase is near zero, is sparse in a renewal sense. The present paper keeps Tao's object and asks a different question: what is the true size of $\max|\chi_n|$, which frequencies attain it, and what mechanism sets its rate? An arXiv search (October 2026) found no previous work that computes $M_n$, identifies the extremal frequencies, or proposes an exponential rate with an explicit constant. Tao's method has been extended to generalized maps $x\mapsto(qx+r)/p$ [6]. Siegel and O'Dwyer [7] extend almost-boundedness to Collatz-type maps on rings of integers of number fields by archimedean methods that, unlike Tao's argument, need no Fourier decay estimate for the generalized Syracuse variables. This agrees with our audit of Tao's argument (§6, Scope): Fourier decay is not the binding constraint there. Rees [8] formulates 2-adic equidistribution conjectures for push-forwards of geometric distributions; these are a different (2-adic, not 3-adic) object.

Large-deviation constants for the geometric walk $s_m$ are classical (Cramér; see [4]). The same "neutral-growth" slope $s_m\approx\lambda m$ appears in heuristic and stochastic models of Collatz trajectories and cycles [2,3]. Here it appears for a different reason: it is the slope along which the phases of the cold walk stay real numbers in $[0,1)$ rather than equidistributed residues. Renewal arguments for Fourier decay of self-similar and stationary measures [5] are the closest methodological analogue. There, renewal theory gives polynomial or logarithmic rates unless extra Diophantine input is available, which matches the dichotomy we find (Section 6).

3. Methods and proofs

3.1 Recursion and exact algorithm

Write $\psi_n(t)=\chi_n(2^t)$ for $t\in\mathbb{Z}/N_n$, with $N_n=2\cdot3^{n-1}$. Peel off the first digit: $\mathrm{Syrac}_n\overset{d}{=}2^{-a}(1+3Y')$, where $Y'\sim\mathrm{Syrac}_{n-1}$ is independent of $a$. Since $e(2^{t-a}\cdot 3Y'/3^n)=e(2^{t-a}Y'/3^{n-1})$, this gives

$$ \psi_n(t)=\sum_{a\ge1}2^{-a}\,e\!\left(\frac{2^{t-a}}{3^n}\right)\psi_{n-1}(t-a),\qquad \psi_0\equiv1 . $$

The operator $Kf(t)=\sum_{a\ge1}2^{-a}f(t-a)$ on $\mathbb{Z}/N$ satisfies $Kf(t)=\tfrac12\bigl(f(t-1)+Kf(t-1)\bigr)$. It can therefore be applied in $O(N)$ operations: one closed-form initial value from the geometric series, then a linear sweep. All units at level $n$ thus cost $O(3^n)$, and all levels up to $n$ cost $O(3^n)$ in total. We ran this in double precision for $n\le18$ ($N_{18}\approx2.6\times10^8$). We checked it against an independent computation of the exact laws for $n\le 9$; the agreement was $5\times10^{-10}$, limited by the printed precision of the profile.

3.2 The renewal identity

Fix $j\ge0$ and $u$ coprime to $3$. Let $\tau=\min\{m\ge1:\ s_m>j\}$; if no $m\le n$ qualifies, $\tau\geq n+1$. Write $\ell=n-\tau+1$, which is $\leq 0$ when $\tau>n$, and set $\chi_\ell\equiv1$ for $\ell\le0$.

Theorem 1 (renewal identity). For every $n\ge1$, $j\ge0$ and $3\nmid u$,

$$ \chi_n(2^j u)=\mathbb{E}\Bigl[e\Bigl(u\sum_{m<\min(\tau,n+1)}\frac{2^{\,j-s_m}}{3^{\,n-m+1}}\Bigr)\,\chi_{n-\tau+1}(u)\Bigr] =\mathcal{M}_n(j,u)+\sum_{\ell=1}^{n}W_\ell(j,u)\,\chi_\ell(u), $$

where $\mathcal{M}_n(j,u)=\mathbb{E}\bigl[e(u\sum_{m\le n}2^{j-s_m}/3^{n-m+1});\ \tau>n\bigr]$ and $W_\ell(j,u)=\mathbb{E}\bigl[e(u\sum_{m<\tau}2^{j-s_m}/3^{n-m+1});\ \tau=n-\ell+1\bigr]$. In particular,

$$ |\mathcal{M}_n(j,u)|\le\mathbb{P}(s_n\le j),\qquad |W_\ell(j,u)|\le\mathbb{P}(\tau=n-\ell+1)\le\mathbb{P}(s_{n-\ell}\le j). $$

Proof. In forward indexing, $\chi_n(\xi)=\mathbb{E}\,e\bigl(\sum_{m=1}^n\xi\,2^{-s_m}/3^{n-m+1}\bigr)$; here $2^{-s}$ means the inverse modulo $3^{n-m+1}$, which is legitimate because $3^{m-1}$ divides the $m$-th term. Put $\xi=2^ju$. For $m<\tau$ we have $s_m\le j$, so the $m$-th term is the rational number $u2^{j-s_m}/3^{n-m+1}$ with an integer numerator. Now condition on $\mathcal{F}_{\tau-1}=\sigma(a_1,\dots,a_{\tau-1})$ and on the event $\{\tau=m\}=\{s_{m-1}\le j<s_m\}$. By memorylessness, the overshoot $b_1=s_\tau-j$ is again $\mathrm{Geom}(1/2)$ on $\{1,2,\dots\}$ and is independent of $\mathcal{F}_{\tau-1}$. The digits $a_{\tau+1},a_{\tau+2},\ldots$ are i.i.d. and independent of both. With $b_i=s_{\tau+i-1}-j$, the terms $m\ge\tau$ add up to

$$ u\sum_{i=1}^{\ell}\frac{2^{-b_i}}{3^{\ell-i+1}}\ \overset{d}{=}\ \frac{u\,\mathrm{Syrac}_\ell}{3^\ell}\quad(\ell=n-\tau+1). $$

Its conditional expectation is $\chi_\ell(u)$. The bounds follow from $|e(\cdot)|=1$, $|\chi_\ell|\le 1$ and $\{\tau=n-\ell+1\}\subseteq\{s_{n-\ell}\le j\}$. $\square$

The identity was verified exactly (error $\le 10^{-16}$) for $(n,j,u)\in\{(8,5,1),(8,10,5),(9,7,-7),(9,12,11),(7,3,25),(9,14,1)\}$ against the exact laws (supplement: check_general_renewal.py).

Computing the weights. For $u=1$, $\mathcal{M}_n$ and all $W_\ell$ come from one forward pass over the cold region $\{(m,s):s\le j\}$. Let $V_m(s)=\mathbb{E}[\,e(\cdot)\,;\ s_1,\ldots,s_m\le j,\ s_m=s]$. Then

$$ W_{n-m+1}=\sum_{s\le j}V_{m-1}(s)\,2^{-(j-s)},\qquad V_m(s)=e\!\left(\frac{2^{j-s}\bmod 3^{n-m+1}}{3^{n-m+1}}\right)\sum_{a\geq 1}2^{-a}V_{m-1}(s-a). $$

The cost is $O(nj)$ big-integer modular powers, which is feasible for $n\le 500$ at $j\approx\lambda n$. Combined with exact $\chi_\ell(1)$ for $\ell\le18$, Theorem 1 reproduces the exact $\chi_n(2^j)$ for $n\le18$ to $10^{-11}$.

3.3 Conditional transfer theorem

Let $\varphi(\theta)=\mathbb{E}e^{-\theta a}=1/(2e^\theta-1)$. Chernoff's inequality gives $\mathbb{P}(s_m\le j)\le e^{\theta j}\varphi(\theta)^m$ for every $\theta>0$. Choose

$$ e^{\theta^*}=\frac{\lambda}{2(\lambda-1)},\qquad\text{so that}\qquad \varphi(\theta^*)=\lambda-1\approx0.58496 . $$

Theorem 2. Suppose $|\chi_\ell(1)|\le B\rho^\ell$ for all $\ell\ge1$, where $\rho<\lambda-1$. Then for all $n\ge1$ and $j\ge0$,

$$ |\chi_n(2^j)|\ \le\ \Bigl(1+\frac{Bq}{1-q}\Bigr)\,e^{\theta^* j}(\lambda-1)^n,\qquad q=\frac{\rho}{\lambda-1}. $$

Writing $j=\lambda n+d$, the right-hand side is $C_{B,q}\,\exp(-In+\theta^*d)$ with $I=\lambda(\ln2-H(1/\lambda))=0.054980$. Hence the bound is exponentially small whenever $j\le(\lambda+I/\theta^*)n-\omega(1)\approx1.766\,n$.

Proof. By Theorem 1,

$$ |\chi_n(2^j)|\le\mathbb{P}(s_n\le j)+\sum_{\ell\ge1}B\rho^\ell\,\mathbb{P}(s_{n-\ell}\le j)\le e^{\theta^*j}\Bigl[(\lambda-1)^n+B\sum_{\ell\ge1}\rho^\ell(\lambda-1)^{n-\ell}\Bigr]. $$

For the exponent, use $\lambda H(1/\lambda)=\lambda\ln\lambda-(\lambda-1)\ln(\lambda-1)$, which gives $\theta^*\lambda+\ln(\lambda-1)=\lambda\ln\lambda-\lambda\ln2-(\lambda-1)\ln(\lambda-1)=-I$. $\square$

Remarks. (i) $I$ is exactly the Cramér rate of $\mathbb{P}(s_n\le\lambda n)$. With $\theta^*$ optimal at slope $\lambda$, the main term alone is at most $e^{-In}$. (ii) The same argument with $\chi_\ell(u)$ in place of $\chi_\ell(1)$ treats the frequencies $2^ju$ for any fixed $u$. (iii) The hypothesis is the crux. The renewal identity at $j=0$ is trivial ($\tau=1$), so Theorem 1 cannot bootstrap it. Under square-root cancellation, $|\chi_\ell(1)|\approx3^{-\ell/2}=0.5774^\ell$ and $q\approx 0.987$: the hypothesis would hold, but only by a margin of $0.013$ nats per level. Exact data show slightly more cancellation, since typical units decay like $e^{-0.564\ell}$ (§4.3), and the margin is then about $0.028$. Section 4.3 certifies the hypothesis rigorously for $\ell\le800$.

Corollary 3 (computer-assisted). For all $1\le n\le1000$ and all $j\ge0$,

$$ |\chi_n(2^j)|\ \le\ 39\,e^{\theta^*j}(\lambda-1)^n;\qquad\text{in particular}\qquad |\chi_n(2^j)|\le 39\,e^{-In+\theta^*(j-\lambda n)}. $$

Proof. In the proof of Theorem 2, $\chi_\ell(1)$ enters only for $1\le\ell\le n$. Section 4.3 certifies $|\chi_\ell(1)|\le B\rho^\ell$ for $1\le\ell\le1000$ with $\rho=0.58$ and $B=5.6961$, using rigorous ball arithmetic and an exact bound on the truncated tail. The proof of Theorem 2 gives the constant $1+\sum_{\ell=1}^{n}|\chi_\ell(1)|(\lambda-1)^{-\ell}$. Summing the certified upper bounds for $\ell\le1000$ (all terms are nonnegative) gives $1+\sum_{\ell\le1000}|\chi_\ell(1)|(\lambda-1)^{-\ell}\le38.95$ (cert_constant.py). (The uniform bound $B\rho^\ell$ alone gives $q=0.58/(\lambda-1)=0.99152\ldots$ and $1+Bq/(1-q)\le666.8$.) $\square$

This is nontrivial (below $1$) when $In-\theta^*(j-\lambda n)>\ln39=3.66$, for example for $j\le\lambda n$ and $n\ge67$. For general frequencies it says nothing; for those see §5.

3.4 Exact evaluation at a single frequency in polynomial time

The recursion in §3.1 costs $O(3^n)$ because it produces all units at once. A single coefficient is much cheaper. By the forward formula,

$$ \chi_n(2^ju)=\mathbb{E}\prod_{m=1}^{n}e\!\left(\frac{u\,2^{\,j-s_m}\bmod 3^{n-m+1}}{3^{n-m+1}}\right), $$

where negative exponents are inverses modulo $3^{n-m+1}$. Each factor depends only on $(m,s_m)$, so with $V_m(s)=\mathbb{E}[\prod_{i\le m}(\cdots);\ s_m=s]$ we have

$$ V_m(s)=e\!\left(\frac{u2^{j-s}\bmod 3^{n-m+1}}{3^{n-m+1}}\right)\sum_{a\ge1}2^{-a}V_{m-1}(s-a),\qquad \chi_n(2^ju)=\sum_s V_n(s). $$

The convolution again obeys $G(s)=\tfrac12(V(s-1)+G(s-1))$. We truncate at $s\le S_{\max}=2n+12\sqrt{2n}+60$; the discarded mass is at most $\mathbb{P}(s_n>S_{\max})<10^{-25}$ by Chernoff. The cost is $O(nS_{\max})=O(n^2)$ big-integer multiplications modulo $3^k$, and phases are rounded to 53 bits from exact integer division. Against the exact tables ($n\le18$) the agreement is $5\times10^{-11}$, limited by the printed precision. An 80-bit recomputation agrees to relative error $10^{-19}$ for $\ell\le50$, because the entries of $V_m$ shrink together with the result. This algorithm supersedes the Monte Carlo and hybrid computations (§§4.4–4.5), which we keep only as independent cross-checks.

4. Results and supporting evidence

4.1 Exact values, $n\le18$

| $n$ | $M_n$ | $\mathbb{E}_u|\chi_n(u)|^2$ | $n$ | $M_n$ | $\mathbb{E}_u|\chi_n(u)|^2$ | |---|---|---|---|---|---| | 1 | 5.774e-1 | 3.33e-1 | 10 | 3.828e-2 | 1.19e-5 | | 2 | 3.779e-1 | 7.94e-2 | 11 | 3.194e-2 | 3.96e-6 | | 3 | 2.522e-1 | 2.56e-2 | 12 | 2.646e-2 | 1.32e-6 | | 4 | 1.770e-1 | 8.60e-3 | 13 | 2.205e-2 | 4.42e-7 | | 5 | 1.293e-1 | 2.87e-3 | 14 | 1.913e-2 | 1.47e-7 | | 6 | 9.611e-2 | 9.59e-4 | 15 | 1.628e-2 | 4.92e-8 | | 7 | 7.587e-2 | 3.19e-4 | 16 | 1.441e-2 | 1.64e-8 | | 8 | 6.089e-2 | 1.07e-4 | 17 | 1.251e-2 | 5.48e-9 | | 9 | 4.803e-2 | 3.56e-5 | 18 | 1.119e-2 | 1.83e-9 |

For $8\le n\le18$, the maximum is attained at $\xi=2^{j^*}$ with $j^*\in\{9,10,12,13,14,16,17,18,20,21,23\}$, that is, $j^*=\lambda n-c$ with $c\in[3.6,5.6]$. For $n \le 7$ the arg-max is the conjugate representative $-2^{j}$. For $n\le18$ the mean square over units lies between $0.69$ and $1.00$ times $3^{-n}$. By Parseval it equals $(3^n\mathrm{CP}_n-3^{n-1}\mathrm{CP}_{n-1})/(2\cdot3^{n-1})$, where $\mathrm{CP}_n=\sum_x\mathbb{P}(\mathrm{Syrac}_n=x)^2$ is the collision probability. Every one of the twelve largest units at each level has the form $\pm2^{j}(1+3^{n-1}t)$, which is a lift of a level-$(n-1)$ extremizer. Over these 18 levels, $M_n$ falls by a factor of only about $50$. The local rate $\ln(M_{17}/M_{18})=0.11$ is far below the generic rate $\tfrac12\ln3=0.55$.

4.2 Exact values along the extremal family, $n\le2000$

With the single-frequency algorithm of §3.4 we computed $M^*_n=\max\{|\chi_n(2^j)|:\ \lambda n-14\le j\le\lambda n+3\}$. For $n\le18$ this window contains the global maximum $M_n$. For larger $n$ it is a lower bound for $M_n$, and we conjecture equality; exact multiplier scans at $n=40,80,150$ (below) and a Monte Carlo scan at $n=30$ (§4.4) support this.

| $n$ | $j^*$ | $j^*-\lambda n$ | $M^*_n$ | $n\,e^{In}M^*_n$ | |---|---|---|---|---| | 20 | 26 | -5.69 | 8.885e-03 | 0.534 | | 30 | 42 | -5.54 | 3.288e-03 | 0.513 | | 44 | 64 | -5.73 | 9.893e-04 | 0.489 | | 62 | 92 | -6.26 | 2.435e-04 | 0.456 | | 80 | 120 | -6.79 | 6.453e-05 | 0.420 | | 100 | 152 | -6.49 | 1.594e-05 | 0.389 | | 120 | 184 | -6.19 | 4.120e-06 | 0.363 | | 150 | 231 | -6.74 | 5.964e-07 | 0.341 | | 180 | 279 | -6.29 | 9.177e-08 | 0.328 | | 220 | 342 | -6.69 | 8.232e-09 | 0.324 | | 260 | 405 | -7.09 | 7.681e-10 | 0.322 | | 300 | 469 | -6.48 | 7.425e-11 | 0.324 | | 400 | 627 | -6.98 | 2.302e-13 | 0.327 | | 500 | 786 | -6.48 | 7.553e-16 | 0.328 | | 650 | 1023 | -7.22 | 1.497e-19 | 0.322 | | 800 | 1261 | -6.97 | 3.130e-23 | 0.317 | | 1000 | 1578 | -6.96 | 4.033e-28 | 0.304 | | 1200 | 1895 | -6.96 | 5.391e-33 | 0.291 | | 1400 | 2212 | -6.95 | 7.411e-38 | 0.278 | | 1700 | 2688 | -6.44 | 3.920e-45 | 0.260 | | 2000 | 3163 | -6.93 | 2.159e-52 | 0.245 |

For $n\ge1200$ the window is $[\lambda n-9,\lambda n-4]$. It contains $j^*$ for every $n\ge62$ in the table. At $n=1000$ an 80-bit recomputation agrees with the float64 value to 7 printed digits ($4.032640\times10^{-28}$). Least-squares fits of $\log M^*_n=a-\beta\log n-\kappa n$ give $(\beta,\kappa)=(0.88,0.05529)$ for $n\ge300$, $(0.92,0.05525)$ for $n\ge500$ and $(1.02,0.05518)$ for $n\ge800$, against $I=0.054980$. With $\kappa=I$ fixed, $\beta=1.15$, $1.21$ and $1.28$ on the same ranges, and the two-point exponents are $1.26$ on $[1000,1400]$ and $1.35$ on $[1400,2000]$. So the rate is $I$ to within $0.5\%$, but the polynomial correction has not converged. The plateau $n\,e^{In}M^*_n\approx0.32$ for $180\le n\le800$ is pre-asymptotic. Adding a correction term, $\log(e^{In}M^*_n)=a-\beta\log n+b\,n^{-1/2}$, does not stabilise the estimate either: the fit gives $\beta=1.17\pm0.07$, $1.48\pm0.05$, $1.67\pm0.03$ and $1.79\pm0.02$ for $n\ge100,180,300,500$ (formal errors). We therefore regard $\beta$ as undetermined; the bridge value $3/2$ of §5 is plausible but not established. The normalized column also wobbles because $j$ is an integer: $j^*-\lambda n$ moves through $[-7.2,-6.2]$ as $\{\lambda n\}$ varies. The local rate decreases monotonically, from $0.099$ ($n=20$–$30$) to $0.0558$ ($n=1400$–$2000$).

Certified two-sided bounds. With the ball-arithmetic DP of §4.3 (certify_M.py) we certified the table entries at $(n,j^*)=(100,152),(200,310),(400,627),(600,944),(800,1261),(1000,1578)$. This gives rigorous lower bounds $M_n\ge1.5936\times10^{-5}$, $2.7273\times10^{-8}$, $2.3022\times10^{-13}$, $2.5620\times10^{-18}$, $3.1301\times10^{-23}$ and $4.0326\times10^{-28}$, which agree with the float64 values to all printed digits. At the same $(n,j^*)$, Corollary 3 gives the rigorous upper bounds $2.2\times10^{-2}$, $7.8\times10^{-5}$, $1.3\times10^{-9}$, $2.2\times10^{-14}$, $3.7\times10^{-19}$ and $6.2\times10^{-24}$. The ratio of upper to lower bound grows only from $1.4\times10^3$ to $1.5\times10^4$, roughly linearly in $n$, while both decay through 23 orders of magnitude. So at these six points the true size along the extremal family is rigorously bracketed within a factor of at most $1.6\times10^4$ of $39\,e^{-In+\theta^*(j^*-\lambda n)}$. This is a finite statement, and it is consistent with C1.

Is $2^j$ the extremal family for $n>18$? We scanned all $|\chi_n(2^ju)|$ exactly with $1\le u\le U$, $u$ odd, $3\nmid u$, and $0\le j\le J$, using $(n,U,J)=(40,199,120)$, $(80,199,160)$ and $(150,101,260)$. That is 8107, 10787 and 8874 frequencies; since $|\chi_n(-\xi)|=|\chi_n(\xi)|$, signs are covered. We also scanned 3000, 2000 and 500 uniformly random units. In every case $u=1$ attains the maximum, at the same $j^*$ as in Table 2. The best competitor is always $u=13=2^4-3$, at $j=j^*-3$ or $j^*-4$, with ratio $0.33$, $0.33$ and $0.38$. This is a second family with the same exponential rate, consistent with the near-relation picture of §5. The random units have root-mean-square size $0.72$, $0.64$ and $0.57$ times $3^{-n/2}$, and their maxima are $10$ to $20$ times their root-mean-square size. They lie more than $10^{28}$ below $M^*_{150}$.

4.3 The renewal input $\chi_\ell(1)$, $\ell\le800$

The same algorithm gives $\chi_\ell(1)$ for every $\ell\le800$ (chi1_scan.txt, chi1_scan_401_800.txt). A linear fit of $\ln|\chi_\ell(1)|$ over $50\le\ell\le800$ has slope $-0.565$, with residual standard deviation $2.0$ (the values fluctuate like a typical random sum). The constants $B(\rho)=\max_{\ell\le800}|\chi_\ell(1)|\rho^{-\ell}$ are $B(0.575)=17.7$, $B(0.578)=8.96$, $B(0.580)=5.70$ and $B(0.584)=2.32$, all attained at $\ell=131$; for $401\le\ell\le800$ every value satisfies $|\chi_\ell(1)|\le e^{-2.8}\,3^{-\ell/2}$. For comparison we drew 20–30 uniformly random units $u$ at each of $\ell=50,100,200,\dots,800$ (rand_units.txt). The median of $\ln(|\chi_\ell(u)|\,3^{\ell/2})$ is $-1.5,-2.5,-3.9,-5.8,-6.2,-7.3,-9.9,-11.5,-12.1$, which is linear in $\ell$ with slope $-0.014$. So typical units decay at rate $\approx0.564$, slightly faster than square-root cancellation ($0.549$). The spread of $\ln|\chi_\ell(u)|$ also grows, with the sample range rising from $3.9$ at $\ell=50$ to $7.7$ at $\ell=800$. So the median rate and the root-mean-square rate can differ: by Jensen, $\mathbb{E}_u\ln|\chi_\ell(u)|\le\tfrac12\ln\mathbb{E}_u|\chi_\ell(u)|^2$, so a typical ("quenched") rate above $\tfrac12\ln3$ is compatible with a mean square of order $3^{-\ell}$, which would then be carried by rare large units. Our samples are too small to estimate the mean square at large $\ell$. The block medians of $\ln(|\chi_\ell(1)|\,3^{\ell/2})$ over $\ell\in[1,100],\dots,[701,800]$ track them, from $-1.0$ to $-13.4$, so $u=1$ is a typical unit. The hypothesis of Theorem 2 is therefore the statement that one fixed frequency shows the typical cancellation. A phase-perturbation test (sensitivity.py) multiplies every phase factor by $e^{i\epsilon g}$ with $\epsilon=10^{-10}$ and $g$ standard normal. It changes $\chi_{800}(1)\approx2.0\times10^{-197}$ by a relative $4.6\times10^{-9}$, so rounding amplification is below $50$ and the float64 values are reliable despite their size.

Rigorous certification. certify_chi1.py repeats the DP for $u=1$, $j=0$ and every $\ell\le1000$ in Arb ball arithmetic (python-flint 0.6.0). Each phase $e(x/3^k)$ is enclosed from the exact rational $x/3^k$. Entries with $s_m>S$ are dropped, and since $s_m$ is increasing, the dropped contribution has modulus at most $\mathbb{P}(s_\ell>S)=\mathbb{P}(\mathrm{Bin}(S,\tfrac12)\le\ell-1)$, which we compute exactly as a rational. We take the smallest $S\in2\ell+16\mathbb{Z}_{\ge0}$ with tail at most $2^{-0.786\ell-20}$. Rectangular complex balls inflate by up to $|\cos\theta|+|\sin\theta|$ (on average $4/\pi$) at every phase multiplication, so a working precision of only $0.8\ell$ bits loses all accuracy by $\ell\approx800$. With $1.25\ell+100$ bits the final ball radius is below $10^{-28}$ relative to the value. The resulting rigorous upper bounds (certify_chi1.txt; about 38 min for $\ell\le800$ and 52 min for $801\le\ell\le1000$, with 5 processes) give $\max_{\ell\le1000}\overline{|\chi_\ell(1)|}\,0.58^{-\ell}=5.6961$ at $\ell=131$. The certified bounds agree with the float64 values to relative error $5\times10^{-10}$, which confirms the float computations independently.

4.4 Cross-check: Monte Carlo, $20\le n\le62$

Each run uses $S=2\times10^8$ samples of $(a_1,\dots,a_n)$ and exact 128-bit arithmetic mod $3^n$, and evaluates $\chi_n(2^j)$ for $0\le j<J$. The standard error is about $5\times10^{-5}$ per component.

| $n$ | $\max_j|\hat\chi_n(2^j)|$ (MC) | $j^*$ | $\lambda n-j^*$ | exact $|\chi_n(2^{j^*})|$ (§3.4) | |---|---|---|---|---| | 20 | 8.84e-3 | 26 | 5.7 | 8.885e-3 | | 24 | 5.82e-3 | 32 | 6.0 | 5.831e-3 | | 30 | 3.19e-3 | 41 | 6.5 | 3.254e-3 | | 36 | 1.89e-3 | 51 | 6.1 | 1.950e-3 | | 44 | 1.03e-3 | 64 | 5.7 | 9.893e-4 | | 52 | 5.70e-4 | 76 | 6.4 | 5.253e-4 | | 62 | 2.73e-4 | 92 | 6.3 | 2.435e-4 |

A multiplier scan at $n=30$ over $\chi_{30}(c\,2^j)$ for small odd $c$ confirms that $c=1$ dominates ($3.2\times10^{-3}$; the next largest is $c=11$ at $1.25\times10^{-3}$). The Monte Carlo maxima agree with the exact values to within about $1.5$ standard errors, except at $n=62$, where taking the maximum over $J$ noisy values biases the estimate upward by $4\sigma$. Fitted on $20\le n\le62$ alone, the form $\log M_n=a-\beta\log n-\kappa n$ gives $\beta\approx1.31$ and $\kappa\approx0.047$, both far from the large-$n$ values of §4.2: in this range the data are pre-asymptotic.

4.5 Cross-check: renewal decomposition at large $n$

Theorem 1 can be evaluated independently of §3.4. The weights $\mathcal{M}_n,W_\ell$ come from the cold-walk recursion (§3.2) and $\chi_\ell(1)$ from §4.3. At $(n,j)=(100,152)$, $(200,310)$ and $(300,469)$ the two computations agree to relative error $2\times10^{-11}$, $5\times10^{-11}$ and $6\times10^{-11}$. The decomposition also shows where the mass sits. The main term is only $0.9\%$, $0.6\%$ and $0.5\%$ of $|\chi_n(2^j)|$, and the largest single renewal term has $\ell=7$, $17$ and $17$. So the extremal coefficients are governed by the renewal sum, i.e. by the near-critical competition in Theorem 2. They are not governed by the cold-walk probability $\mathbb{P}(s_n\le j)$ alone.

An earlier hybrid model replaced $\chi_\ell(1)$, $\ell>18$, by independent random phases with variance $\mathbb{E}_u|\chi_\ell(u)|^2$. It reproduces $M^*_n$ to within $10\%$ for $n\le300$ and recovers the rate $I$, but it overestimates the large-$n$ prefactor: it predicts a local $\beta\approx0.4$, whereas the exact data give $1.0$–$1.35$ (§4.2). We keep it only as evidence that the rate does not depend on the arithmetic details of $\chi_\ell(1)$. Using only $|\chi_\ell(1)|\le1$ for $\ell>18$ certifies $M^*_{20}$ and $M^*_{24}$ to within $1\%$; beyond $n\approx30$ that crude certificate fails.

4.6 Shape of the extremal profile

The exact profile $|\chi_n(2^j)|/M^*_n$ has the same shape at every $n$ (profile_exact.txt):

| $j-j^*$ | $-8$ | $-6$ | $-4$ | $-2$ | $0$ | $+2$ | $+4$ | $+6$ | $+8$ | $+10$ | |---|---|---|---|---|---|---|---|---|---|---| | $n=62$ ($j^*-\lambda n=-6.3$) | 0.212 | 0.385 | 0.627 | 0.880 | 1.000 | 0.813 | 0.375 | 0.078 | 0.024 | 0.004 | | $n=150$ ($j^*-\lambda n=-6.7$) | 0.286 | 0.453 | 0.668 | 0.886 | 1.000 | 0.862 | 0.461 | 0.110 | 0.024 | 0.004 | | $n=400$ ($j^*-\lambda n=-7.0$) | 0.307 | 0.468 | 0.670 | 0.878 | 1.000 | 0.892 | 0.513 | 0.140 | 0.032 | 0.014 |

On the left, the logarithmic slope grows with distance from the peak, reaching $0.33$ ($n=62$), $0.25$ ($n=150$) and $0.23$ ($n=400$) nats per unit of $j$ at $j^*-9$. The upper bound of Theorem 2 has slope $\theta^*=0.304$: a smaller $j$ requires a larger deviation for the walk to stay cold, but the renewal terms soften the penalty. On the right the profile falls about $250$-fold over $10$ units of $j$ (per-unit factors between $1.2$ and $3$), because the walk now starts in the hot-archimedean region $r>\lambda k$, where phases $2^{r}/3^{k}$ are large real numbers. Those first rows cancel through the Fourier transform of the real stationary law $Z=\sum_{m\ge1}3^{m-1}2^{-s_m}$, which satisfies $Z\overset{d}{=}2^{-a}(1+3Z')$. A Monte Carlo estimate ($2\times10^7$ samples) gives $|\widehat{Z}(2^t)|\approx 0.037\cdot 2^{-0.58t}$ for integer $0\le t\le 10$ (half-integer $t$ are smaller), a power decay at an exponent consistent with an absolutely continuous stationary measure. The maximum, at $d=j^*-\lambda n\in[-7.2,-6.2]$ for all $62\le n\le2000$ in Table 2, is the balance point between these two effects. It stays bounded as $n$ grows, which is why the extremal frequencies are $2^{\lambda n-O(1)}$.

5. Conjecture and the cold-walk heuristic

Conjecture C1. $\displaystyle\lim_{n\to\infty}\frac1n\log\max_{3\nmid\xi}|\chi_n(\xi)|=-I,\qquad I=\lambda\bigl(\ln2-H(1/\lambda)\bigr)\approx0.05498,$ with near-extremal frequencies $\xi=\pm2^{\lambda n+O(1)}$. The polynomial correction is not settled. Ballot-type heuristics for a walk constrained near the line $s_m\approx\lambda m$ suggest $n^{-3/2}$, and the main term $\mathcal{M}_n$ alone has local exponent $1.6$ at $n=500$. However, the renewal sum, which dominates (§4.5), has a near-critical geometric weight $(|\chi_\ell(1)|/(\lambda-1)^\ell)$. In the exact data the effective exponent (with $\kappa=I$ fixed) is $1.0$ on $180\le n\le800$, $1.26$ on $[1000,1400]$ and $1.35$ on $[1400,2000]$. It drifts upward, and fits with a correction term do not stabilise (§4.2), so the exponent is undetermined.

Why this rate. Write $k=n-m+1$ for the level and $r=j-s_m$ for the column. For $\xi=2^j$ the phase of the $m$-th term is the real number $2^r/3^k$ whenever $r\ge0$. It is tiny in the cold triangle $0\le r\lesssim\lambda k$, where it contributes no cancellation; it is effectively uniform when $r>\lambda k$; and when $r<0$ it is a 2-adic residue, which is where the renewal terms of Theorem 1 come from. A walk with mean slope $2>\lambda$ that starts at $r=\lambda n$ crosses $r=0$ after about $\lambda n/2$ steps, so about $79\%$ of the rows are cold at no cost. To avoid cancellation entirely, the walk must stay inside the cold triangle for all $n$ rows. This is the event $s_m\le\lambda m+O(1)$, which has probability $e^{-In+o(n)}$ by Cramér's theorem. These are exactly the walks with neutral growth $2^{s_m}\approx3^m$, the regime of heuristic Collatz cycles. Leaving the triangle early costs cancellation at a rate of roughly $0.56$ per row in the renewal part, the typical decay rate of $\chi_\ell$ found in §4.3. Theorem 2 shows that this trade-off is won by staying cold, provided $\chi_\ell(1)$ decays faster than $(\lambda-1)^\ell$.

Comparison with the random-map heuristic. A random function on $\approx4^{n}$ inputs with values in $\mathbb{Z}/3^n$ would have Fourier coefficients of root-mean-square size $3^{-n/2}$. The exact Syracuse law has the same mean square for $n\le18$ (§4.1), and at large $\ell$ its typical units decay slightly faster, at rate $0.564$ (§4.3). For a random-map law $\max_u|\chi_n(u)|$ would decay like $3^{-n/2}$ up to a polynomial factor, i.e. at rate $\tfrac12\ln3=0.549$. C1 instead predicts a rate ten times smaller. So, measured by the median, the Syracuse law is slightly better than random at typical frequencies, and it is far worse at the extremal ones. Tao's predicted fine-scale mixing $\exp(-cm)$ [1] should therefore hold, but with a constant set by the neutral-growth large deviation rather than by entropy counting. Since $|\chi_n(\xi)|\le\mathrm{Osc}_{n-1,n}$, C1 (if true) forces $c\le I$ in any bound $\mathrm{Osc}_{m,n}\ll e^{-cm}$.

Other frequencies. For a general unit $\xi$, the near-zero phases $\{\xi2^{-s}/3^{k}\}$ form a disjoint union of right triangles in the $(k,r)$ lattice. Each triangle is generated by an apex residue $x$ with $|x|/3^{k}=e^{-L}$. Using $\{3^{-1}\cdot\}$ along levels and $\{2\,\cdot\}$ along columns, the triangle has about $L/\ln3$ rows and $L/\ln2$ columns. A walk of slope $2$ spends at most about $L/(2\ln2)$ rows inside it. Two apexes $x_i$ (level $k_i$) and $x_j$ (level $k_j<k_i$, column shift $d$) satisfy $x_i\equiv 2^{d}x_j\pmod{3^{k_j}}$ up to the lattice maps. If they belong to distinct maximal triangles, then $x_i-2^dx_j=3^{k_j}z$ with $z\neq0$, and hence $2^{d}|x_j|+|x_i|\ge3^{k_j}$. In words, deep triangles cannot be packed closely, so their total black time is bounded by a "digit budget". Each black column costs $\ln2$ of budget while each row supplies $\ln3$, which bounds the black fraction of a typical walk by $\ln3/(2\ln2)\approx0.79$. The bound is attained by $\xi=2^j$. This supports C1 for all units, but it is not a proof. The obstruction is long chains of near-relations $2^a3^b\approx$ small, i.e. Diophantine approximation of $\lambda$ through the 2-adic digits of $3^{-k}$. We do not know how to bound the total length of such chains with only an $o(n)$ loss.

Tao's pairing majorant. Along the extremal family $j\approx\lambda n-3$, Tao's pairing majorant $\Psi_n(j)\ge|\chi_n(2^j)|$, computed exactly, takes the values $0.198, 0.083, 0.0173, 0.00387, 0.000915$ at $n=20,40,80,120,160$. It decays exponentially at the local rate $0.036<I$. For $n\le16$, the global maximum of $\Psi_n$ over all $j$ also occurs at $j\approx\lambda n-3$. The polynomial loss in Tao's theorem must therefore come from the worst-case Diophantine configurations above, not from the extremal family.

6. Limitations and scope

• No unconditional exponential bound for all $n$. Theorem 2 needs $|\chi_\ell(1)|\le B\rho^\ell$ with $\rho<\lambda-1$ for all $\ell$. We certify this rigorously for $\ell\le1000$ ($B(0.58)=5.6961$, Corollary 3), but a finite computation cannot cover all $\ell$. Theorem 1 cannot bootstrap the hypothesis, because the identity is trivial at $j=0$. Also, Theorem 2 covers only $\xi=2^j$ with $j\lesssim1.77n$. For general units we have only the heuristic of §5.
• Why renewal gives polynomial rates (heuristic). For $\xi=2^ju$ with $j\approx\lambda n$, Theorem 1 sends $n$ to a renewal index $\ell\approx0.21n$. If the cold segment provides only a fixed factor $c'<1$ of extra cancellation, iterating $M_n\lesssim c'M_{0.21n}$ gives $M_n\lesssim n^{-\log(1/c')/\log(1/0.21)}$, which is Tao-type polynomial decay. Exponential decay requires the gain to grow linearly with the number of cold rows, and that in turn requires control of the 2-adic Diophantine chains of §5.
• Lower bounds. The only rigorous lower bound we have comes from a point mass. If all $a_i=1$ then $\mathrm{Syrac}_n\equiv(3/2)^n-1\equiv-1\pmod{3^n}$, so $\mathbb{P}(\mathrm{Syrac}_n\equiv-1)\ge2^{-n}$. Fourier inversion, together with $\chi_n(3^{n-k}\eta)=\chi_k(\eta)$, gives $2^{-n}\le3^{-n}\bigl(1+\sum_{k\le n}2\cdot3^{k-1}M_k\bigr)$. Hence $\liminf_k-\frac1k\log M_k\le\ln2$, which is far from $I$. For individual $n\le800$ we have certified lower bounds (§4.2), but these are finite statements. The Parseval quantity $(\mathbb{E}_u|\chi_n(u)|^2)^{1/2}\approx0.84\cdot3^{-n/2}$ is a lower bound for $M_n$ only where it has been computed ($n\le18$). We do not know that it stays of order $3^{-n/2}$, and random units at large $n$ are smaller (§4.3). Proving $M_n\ge e^{-(I+o(1))n}$ would require showing that the cold-walk main term is not cancelled by the renewal sum.
• Extremal family. For $n>18$ we compute $|\chi_n(\xi)|$ exactly only along $\xi=2^j$, $j\in[\lambda n-14,\lambda n+3]$. That this family attains $M_n$ is verified for $n\le18$. For $n=40,80,150$ it is supported by exact scans over about $10^4$ small multipliers plus random units (§4.2), but for larger $n$ it is a conjecture. The values $M^*_n$ are therefore lower bounds for $M_n$. They are exact up to floating-point rounding (not interval-certified) and truncation below $10^{-25}$. By contrast, the hypothesis of Theorem 2 for $\ell\le800$, and hence Corollary 3, is certified rigorously by ball arithmetic (§4.3).
• Scope. All results concern Tao's random variable; none of them improves Tao's Collatz theorem. In Tao's alternate form of the main theorem [1, Thm 3.1], the exceptional set has logarithmic density $O(\log^{-c}N_0)$, and the telescoping over scales makes this essentially the per-scale error $\varepsilon(N_0)$ of the first-passage stabilisation. In that error, the Fourier input enters through fine-scale mixing at $m_0\asymp\log x$ and costs only $O_A(\log^{-A}x)$ for every $A$. The exponent $c$ comes instead from archimedean steps: trimming the passage-time window by $\log^{0.8}x$, the concentration of $a_{[1,n]}$ within $\log^{0.6}x$, and the count of the window. An exponential Fourier bound would make the Fourier term $O(x^{-c'})$ but would leave $c$ unchanged. The value of the present results is the quantitative picture of $\mathrm{Syrac}_n$ (rate, extremisers, mechanism and exact algorithm). Strengthening the density estimate itself would require sharper archimedean renewal estimates, and natural density would require mixing of the full random affine map [1, Remark 1.16]. Consistently, [7] obtains Tao-type almost-boundedness without any Fourier decay input.

7. Reproducibility

The supplement contains all sources and outputs. Python $\ge$3.10 with NumPy is required (and python-flint 0.6.0 for the certificate); the C programs need GCC with OpenMP and __int128.

• chi_dp.py: single-frequency exact DP (§3.4); running it executes a self-test against chi_profile.txt.
• exact_M.py n...: the table in §4.2. It takes about 1 min for all $n\le300$ and about 15 min for $n=1000$ on one core.
• exact_M_par.py D0 D1 n...: the rows $n\ge1200$ (window offsets $-9,-4$; 5 processes; about 18 min in total for $n=1200$–$2000$).
• certify_chi1.py 1 800 and certify_chi1.py 801 1000: the rigorous certificate for Corollary 3 (python-flint 0.6.0; about 38 min with 5 processes); output columns are $\ell$, $S$, midpoint, rigorous upper bound including tail, tail bound, upper bound $/0.58^\ell$ and ball radius.
• certify_M.py n:j ...: rigorous lower bounds at $(n,j^*)$ (§4.2; a few minutes per point).
• cert_constant.py: sums the certified upper bounds into the constant $C\le38.95$ of Corollary 3.
• chi1_scan.py L: $\chi_\ell(1)$ for $\ell\le L$ (§4.3; about 1 min for $L=400$); chi1_scan_par.py 401 800 gives the rest (about 5 min with 5 processes).
• chi_exact.c: the all-units recursion (§4.1; about 12 GB RAM at $n=18$); writes chi_profile.txt.
• check_general_renewal.py, renewal_exact.py: verification of Theorem 1.
• renewal_model.py, profile_model.py: the hybrid model and the profiles (§§4.5–4.6).
• chi_mc.c n S J seed, chi_mc_multi.c: Monte Carlo and the multiplier scan (seed 1, 5 threads).
• pair_bound.c, pair_j.py: Tao's pairing majorant.
• zhat_mc.py: $\widehat Z$.
• rand_units.py R seed l... (seeds 7, 11) and sensitivity.py eps n,j,u ...: random units and the rounding test (§4.3).
• unit_scan.py n U J R seed: the multiplier scan of §4.2 (seeds 1, 2, 3; about 20 s, 70 s and 4 min with 5 processes).

Outputs are in results/.

References

[1] T. Tao. Almost all orbits of the Collatz map attain almost bounded values. Forum of Mathematics, Pi 10 (2022), e12. https://doi.org/10.1017/fmp.2022.8 (arXiv:1909.03562; theorem numbers cited here follow the arXiv version)

[2] J. C. Lagarias. The $3x+1$ problem and its generalizations. American Mathematical Monthly 92 (1985), 3–23. https://doi.org/10.2307/2322189

[3] A. V. Kontorovich and J. C. Lagarias. Stochastic models for the $3x+1$ and $5x+1$ problems. In The Ultimate Challenge: The 3x+1 Problem, AMS, 2010. arXiv:0910.1944

[4] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications, 2nd ed. Springer, 1998. https://doi.org/10.1007/978-1-4612-5320-4

[5] J. Li and T. Sahlsten. Trigonometric series and self-similar sets. Journal of the European Mathematical Society 24 (2022), 341–368. https://doi.org/10.4171/JEMS/1102 (arXiv:1902.00426)

[6] F. Gonçalves, R. Greenfeld and J. Madrid. Generalized Collatz maps with almost bounded orbits. arXiv:2111.06170

[7] M. Siegel and R. O'Dwyer. Almost-boundedness of Collatz-type maps on number rings via archimedean methods. arXiv:2609.33018 (2026)

[8] M. Rees. Push-forward of geometric distributions under Collatz iteration: Part 1. arXiv:2404.12279