A joint formulation of sum-conditioned Syracuse 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 give a joint formulation of earlier sum-conditioned mixing: the fine 3-adic digits of $X_n$ mix even after the exact value of $S_n$ is retained. This statement follows from Allikvere [4, Theorem C] and a geometric projection argument. We also give a direct proof in the joint notation.

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. Allikvere [4, Theorems B and C] establishes sum-conditioned Fourier decay and fine-scale mixing on central exponent slices. His Theorem C covers $0.9n\le m\le n$ and $|s-2n|\le C\sqrt{n\log n}$, with arbitrary polynomial decay. His coarser-scale extension covers $n^{1/4}\le m\le n$. Leen [5, conditioned mixing proposition and Appendix A] also gives central-slice mixing, and uses a timed passage comparison to obtain Gaussian laws.

The joint estimate (1), including its full range $1\le m\le n$, is a corollary of the high-scale part of [4]. It is not a new missing input to Tao's argument. This note records a convenient averaged formulation and a direct proof retaining arbitrary bounded functions of the total sum.

Derivation from [4]. For $0.9n\le m\le n$, disintegrate the joint oscillation over $S_n=s$. On $|s-2n|\le C\sqrt{n\log n}$ use [4, Theorem C]; off that event use the trivial distance bound $2$ and a geometric-sum Chernoff estimate. Choosing $C$ large for each desired exponent gives $O_B(n^{-B})$. For a shorter scale $r$, the projection $(S_n,X_n\bmod3^r)$ has the law of $(S_r+S'_{n-r},X_r)$ with an independent tail sum. Adding this tail is a Markov kernel and contracts the joint distance. Apply the high-scale estimate at length $r$ along a geometric chain of adjacent scales from $n$ to $m$; summing the errors gives $O_B(m^{-B})$. Thus no uniform conditional estimate on very rare slices, or on all smaller moduli within one fixed conditional law, is needed to obtain (1).

Gonçalves, Greenfeld, and Madrid [2, Proposition 4.1] prove offset mixing for a broader class of maps. The calculations in [3] concern conditional characteristic functions of the offset. Allikvere [4] and Leen [5] are public preprints; the comparisons here concern the displayed mixing and passage results, not an independent verification of every result in them.

An integer lift must satisfy both a ternary congruence and an archimedean size constraint. The joint theorem alone does not compare uniform integer samples with logarithmic samples. It 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 direct proof retains the exact exponent sum in the Fourier and Plancherel argument, a mechanism already present in [4]. It 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

[4] Jaan Allikvere. Almost all Collatz orbits attain almost bounded values in natural density, version 2, July 22, 2026. Public preprint, especially Theorems B and C and the coarser-scale proposition in Section 4. https://doi.org/10.5281/zenodo.21499244

[5] David Leen. A Gaussian law for Collatz total stopping times, September 29, 2026. Public preprint, especially the conditioned mixing input, Appendix A, and the timed passage comparison. https://doi.org/10.5281/zenodo.23046896