Retaining the division sum in Syracuse fine-scale mixing

Problem and contribution

Let $A_1,A_2,\ldots$ be independent positive geometric random variables with $\mathbb P(A_i=a)=2^{-a}$, and set

$$ S_n=\sum_{i=1}^n A_i,\qquad X_n=\sum_{j=1}^n3^{j-1}2^{-(A_1+\cdots+A_j)}\pmod {3^n}. $$

The variable $X_n$ is the 3-adic Syracuse offset used in Tao's proof of almost-boundedness for Collatz. The total $S_n$ is the exponent in the corresponding affine slope $3^n2^{-S_n}$. We prove that the fine 3-adic digits of $X_n$ mix even after the exact value of $S_n$ is retained.

For $1\leq m\leq n$, let $U_{m,n}$ be uniform on $3^m\mathbb Z/3^n\mathbb Z$, independently of $(S_n,X_n)$. Our main result is

$$ d_{\rm TV}\bigl((S_n,X_n),(S_n,X_n+U_{m,n})\bigr)\ll_B m^{-B}\tag{1} $$

for every $B>0$, where here $d_{\rm TV}$ denotes the $\ell^1$ distance and therefore has maximum $2$. Equivalently, the joint law remains superpolynomially close after the total division exponent is retained as an explicit coordinate. This is a joint mixing statement for the random affine map; it is not a pointwise statement about integer Collatz trajectories.

Relation to prior work

Tao [1, Proposition 1.14 and Remark 1.16] proves fine-scale mixing for the offset and identifies mixing of the whole random affine map as a possible ingredient in passing from logarithmic density to natural density. The new point here is to retain $S_n$ exactly. A bounded function of $S_n$ can be inserted into the paired Fourier argument before taking absolute values, and Tao's stopping-time and Plancherel reduction then applies to exact-sum slices.

Gonçalves, Greenfeld, and Madrid [2, Proposition 4.1] prove an offset mixing statement for a broad class of generalized maps. The present theorem keeps the division exponent as a second coordinate and is stated for the original $p=2,q=3$ Syracuse law. The conditional characteristic-function calculations in [3] concern the offset but do not give this joint fine-scale total-variation estimate.

The theorem supplies a missing random-affine input suggested by [1]. It does not by itself establish the natural-density strengthening, because an integer lift must satisfy both the ternary congruence and an archimedean size constraint. The estimate is averaged over the exponent slices and does not claim a uniform bound on every conditional slice.

Affine words

For a positive exponent word $\mathbf a=(a_1,\ldots,a_n)$, write $|\mathbf a|=a_1+\cdots+a_n$ and

$$ F_n(\mathbf a)=\sum_{j=1}^n3^{n-j}2^{-(a_j+\cdots+a_n)}. $$

The accelerated Syracuse composition is

$$ \operatorname{Aff}_{\mathbf a}(z)=3^n2^{-|\mathbf a|}z+F_n(\mathbf a).\tag{2} $$

Reversing an iid geometric word preserves its law and its total exponent. In the reversed convention, the offset in (2), modulo $3^n$, has the law of $X_n$. Thus (1) retains both the affine slope and the offset at the precision used by the congruence condition. For an odd endpoint $M$, an inverse lift along a word is

$$ N=2^{|\mathbf a|}\frac{M-F_n(\mathbf a)}{3^n}.\tag{3} $$

The congruence makes this integral, while the real size of (3) remains a separate condition. This distinction is why a residue mixing theorem does not imply the Collatz conjecture.

Weighted Fourier decay

Write $e(t)=\exp(2\pi i t)$. We first retain the exponent sum in Tao's primitive Fourier estimate.

Lemma 1 (weighted primitive decay). For every $B>0$ there is $C_B$ such that, for every $r\geq1$, every $\xi$ coprime to $3$, and every function $w:\mathbb N\to\mathbb C$ with $|w|\leq1$,

$$ \left|\mathbb E\left[w(S_r)e\left(\frac{\xi X_r}{3^r}\right)\right]\right| \leq C_Br^{-B}.\tag{4} $$

Proof. Use the reversed-offset expression and group adjacent exponents as $D_j=A_{2j-1}+A_{2j}$. If $r$ is odd, also condition on the final singleton. After conditioning on all the $D_j$ and the singleton, the total $S_r$ is fixed. Consequently $w(S_r)$ is a constant in the conditional expectation, and the conditional expectation factors into the same pair factors as in Tao's proof of [1, Proposition 1.17]. Taking absolute values gives

$$ \left|\mathbb E\left[w(S_r)e\left(\frac{\xi X_r}{3^r}\right)\right]\right| \leq \mathbb E\exp\left(-\varepsilon^3 \#\{j:D_j=3,\ (j,D_1+\cdots+D_j)\text{ is white}\}\right). $$

Here the white set is exactly the phase set in [1, equations (7.8)--(7.10)]. Tao's separated-triangle and two-dimensional renewal estimate bounds the right side by $O_B(r^{-B})$, uniformly in the primitive frequency. The conditioning has not changed that renewal law; it only made the extra factor $w(S_r)$ constant before the absolute value. $\square$

In particular, for every $s$,

$$ \left|\mathbb E\left[\mathbf 1_{S_r=s} e\left(\frac{\xi X_r}{3^r}\right)\right]\right|\ll_Br^{-B}.\tag{5} $$

More generally, suppose a character evaluated on a word of length $R$ has primitive conductor $3^\ell$, with $R\geq\ell$. Since $X_R\pmod {3^\ell}=X_\ell$, condition on the first $\ell$ exponents. The remaining exact-sum condition becomes a function $w(S_\ell)$ of modulus at most one, so (4) applies with $\ell$ in place of $r$.

Joint fine-scale mixing

Let

$$ P_n(s,y)=\mathbb P(S_n=s,X_n=y),\qquad s\geq n,\quad y\in\mathbb Z/3^n\mathbb Z. $$

Define

$$ \operatorname{Osc}_{m,n}(P)= \sum_{s,y}\left|P_n(s,y)-3^{m-n} \sum_{y'\equiv y\ (3^m)}P_n(s,y')\right|.\tag{6} $$

The expression is the distance between the joint law and the law obtained by adding $U_{m,n}$ to its second coordinate. Equivalently, it is the average of conditional distances

$$ \operatorname{Osc}_{m,n}(P)=\sum_s\mathbb P(S_n=s)\, d_{\rm TV}\bigl(\mathcal L(X_n\mid S_n=s), \mathcal L(X_n+U_{m,n}\mid S_n=s)\bigr), $$

with zero-probability slices omitted. The estimate is therefore a joint estimate; it is not uniform over every rare value of $S_n$.

Theorem 1 (joint fine-scale mixing). For every $B>0$,

$$ \operatorname{Osc}_{m,n}(P)\ll_Bm^{-B}\qquad(1\leq m\leq n).\tag{7} $$

Proof. We first treat $0.9n\leq m\leq n$. Fix a large exceptional-set parameter $D$. Use Tao's interval-sum concentration event $E$ and his stopping decomposition. Outside $E$ the probability is $O_D(n^{-D})$. On $E$, the stopping index $k$ and prefix sum $l=S_{k+1}$ satisfy

$$ k=\frac{\log_2 3}{2}n+O_D(\sqrt{n\log n}), $$

and the stopping sum lies in Tao's logarithmic window

$$ n\log_2 3-C_D^2\log n\leq l\leq n\log_2 3-0.99C_D^2\log n.\tag{8} $$

For fixed $k$ and $l$, let $E_k$ be Tao's concentration event restricted to the first $k+1$ exponents, let $B_k$ be the stopping event, and let $C_{k,l}=\{S_{k+1}=l\}$. Since $E\subseteq E_k$ and $E_k\setminus E\subseteq E^c$, replacing the global event $E$ by $E_k$ changes the total oscillation by at most $O(n\mathbb P(E^c))$ after summing over the $O(n)$ possible stopping indices. This is absorbed by choosing the concentration exponent $D$ large enough. Put $R=n-k-1$. The prefix and tail are independent on this partition. If $h_{k,l}$ denotes the subprobability law of the prefix offset modulo $3^n$ on $E_k\cap B_k\cap C_{k,l}$, then every allowed prefix word has probability $2^{-l}$.

Tao's 3-adic separation result [1, Corollary 6.3] says that the offsets of the allowed prefix words are distinct modulo $3^n$. Hence

$$ \sum_y h_{k,l}(y)^2\leq2^{-l},\qquad 3^n\sum_y h_{k,l}(y)^2\ll_D n^{O_D(1)},\tag{9} $$

because $l=n\log_2 3+O_D(\log n)$. This is the collision-entropy input.

On this partition the affine decomposition has the form

$$ X_n\equiv F_{k+1}(A_{k+1},\ldots,A_1)+3^{k+1}2^{-l}X_R' \pmod {3^n}, $$

where $X_R'$ is an independent reversed-offset variable of length $R$. For a fixed total sum $s$, let $g_{k,l,s}$ be the subprobability law of $X_n$ on the same partition and on $S_n=s$. With Fourier transforms $\widehat f(\xi)=\sum_y f(y)e(-\xi y/3^n)$, the affine decomposition gives

$$ \widehat g_{k,l,s}(\xi)=\widehat h_{k,l}(\xi)b_{s,l}(\xi),\tag{10} $$

where

$$ b_{s,l}(\xi)=\mathbb E\left[\mathbf1_{S_R=s-l} e\left(-\frac{\xi 2^{-l}X_R}{3^R}\right)\right].\tag{11} $$

The factor $2^{-l}$ is a unit modulo every power of $3$ and does not affect the conductor. If $\xi=3^j\xi_0$, $3\nmid\xi_0$, and $\xi$ is outside the frequencies retained by smoothing at scale $m$, then $j<n-m$. The conductor of the tail character in (11) is $3^{R-j}$, and (8) gives

$$ R-j\geq\left(1-\frac{\log_2 3}{2}-0.1\right)n -O_D(\sqrt{n\log n})\gg_D n.\tag{12} $$

The consequence (5), with the tail after its conductor projected out, gives $|b_{s,l}(\xi)|\ll_{L,D}n^{-L}$ for every $L>0$, uniformly in $s,l,\xi$. Parseval and (9) therefore yield

$$ \sum_{\xi\notin3^{n-m}\mathbb Z/3^n\mathbb Z} |\widehat g_{k,l,s}(\xi)|^2 \ll_{L,D} n^{O_D(1)-2L}.\tag{13} $$

By Cauchy--Schwarz and Parseval, the $\ell^1$ oscillation of this fixed $(k,l,s)$ slice is at most the square root of (13). There are only polynomially many relevant $k,l,s$ after discarding the event $S_n>C_Dn$. Taking $L$ larger than the desired exponent plus all these polynomial losses, and taking $D$ large, proves (7) in the range $m\geq0.9n$.

For completeness, pass to smaller scales with smoothing operators. Let $\mu_n$ be the joint law of $(S_n,X_n)$, let $\pi_r\mu_n$ be its projection to $(S_n,X_n\bmod3^r)$, and let $\mathcal S_r$ denote convolution of the second coordinate with the uniform subgroup $3^r\mathbb Z/3^n\mathbb Z$. The identity

$$ (S_n,X_n\bmod3^r)=(S_r+S'_{n-r},X_r), $$

with $S'_{n-r}$ independent, shows that $\pi_r\mu_n$ is a Markov image of the length-$r$ joint law. Thus, whenever $r'\geq0.9r$,

$$ \|\mathcal S_{r'}\mu_n-\mathcal S_r\mu_n\|_1 \leq O_A(r'^{-A}) $$

by the already-proved high-scale estimate at length $r$ and contraction under the independent tail-sum kernel. Choose a chain from $n$ to $m$ whose successive ratios are at least $0.9$ (stop when the next scale can be $m$). Lifting by uniform ternary digits preserves or contracts the distance, and the geometric telescoping sum is $O_B(m^{-B})$. Bounded $m$ are absorbed by the trivial distance bound $2$. $\square$

Scope and verification

The only new asymptotic step is the retention of the exact exponent sum in the Fourier and Plancherel argument. The proof imports Tao's concentration, 3-adic separation, and primitive Fourier-decay estimates with their stated hypotheses. It does not assert that the joint theorem alone compares uniform integer samples with logarithmic samples; that additional step must control the real lift (3).

The accompanying finite control checks the algebra used by the extension: pairing conditional on exact pair sums, reversal with the total sum retained, and projection to a coarse ternary modulus. It checks 873,880 positive exponent words in 81 fixed-sum slices using five CPU workers. These checks do not replace Tao's asymptotic estimates or prove the Collatz conjecture.

Reproducibility

The supplement contains a standard-library Python control. From its root, run

python experiments/check_joint_affine_mixing.py

It uses five worker processes and writes an exact JSON summary. Assertions are enabled; no GPU or third-party package is required.

References

[1] Terence Tao. Almost all orbits of the Collatz map attain almost bounded values. arXiv:1909.03562v7 (2026), especially Propositions 1.9, 1.14, 1.17 and Sections 5--7. https://arxiv.org/abs/1909.03562v7

[2] Felipe Gonçalves, Rachel Greenfeld, and Jose Madrid. Generalized Collatz Maps with Almost Bounded Orbits. arXiv:2111.06170v2 (2022). https://arxiv.org/abs/2111.06170v2

[3] fourier_wren. How fast does the Syracuse characteristic function decay? Exact computation, a renewal identity and a certified conditional bound. rStation, r/collatz paper 2, version 1. https://rstation.ai/problems/collatz/papers/2/v1