REVIEW
The number system in rational base $3/2$ and the $3x+1$ problem
Number Theory · Submitted 18 Apr 2025 · 6 references
Pith's verdict: A clean, honest note that finds a real dictionary between rational base 3/2 words and the Collatz map; worth a proper referee.
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Referee report
The paper
The paper claims
The rational base 3/2 representation of integers is a mirror of the Collatz map: appending digit 1 to an odd number's word computes T(n), and the largest words of length k are exactly the iterates U^k(0).
The reading
What carries the argument
The load-bearing mechanism is the odometer rule for rational base $3/2$ (Proposition 2.3): if $\langle n\rangle=u0v$ with $v\in\{1,2\}^*$, then $\langle n+1\rangle=u2v^\sigma$, where $\sigma$ is the letterwise cyclic map $2\mapsto1\mapsto0\mapsto2$ on the suffix. Combined with the recursive digit construction (Proposition 2.2) giving the last digit of a word, this rule characterizes which one-digit extensions of an admissible word remain admissible and implies that the only way a word can have maximal length for its number is for all its digits to lie in $\{1,2\}$. That last fact, in turn, yields the prefix recursion $\langle\mathrm{sat}(k)\rangle=\langle\mathrm{sat}(k-1)\rangle a_0$ with $a_0\equiv\mathrm{sat}(k-1)\bmod 2$, and therefore the identity $\mathrm{sat}(k)=U(\mathrm{sat}(k-1))$.
Extended reading notes
Core claim
On the paper's own terms, the discovery is that the syntactically natural operations in the rational base $3/2$ numeration system coincide with the Collatz function. If $w=\langle n\rangle$, then exactly one of the one-digit extensions $w0,w1,w2$ is admissible according to the parity of $n$; when $n$ is odd, $w1=\langle T(n)\rangle$. The paper then defines saturated numbers $\mathrm{sat}(k)$ as the largest integers whose words have length $k$, proves that $\langle\mathrm{sat}(k)\rangle$ is the unique length-$k$ word over $\{1,2\}$ in the language, and shows by prefix recursion that $\mathrm{sat}(k)=U(\mathrm{sat}(k-1))$, hence $\mathrm{sat}(k)=U^k(0)$ for all $k\ge1$, with $U(n)=T(n)$ on odd $n$ and $U(n)=T(n)+n+1$ on even $n$. This makes the infinite sequence of longest words an orbit of a map closely related to $T$, and leads the paper to conjecture that the limiting frequency of digit $1$ in these words is $1/2$ and that the infinite limit word is normal.
Load-bearing premise
The proof borrows its odometer rule, the recipe for how the base-$3/2$ word changes when a number is increased by one, from the foundational study of this numeration system and does not prove it again; every main identity, including $\mathrm{sat}(k)=U^k(0)$, is a corollary of that rule, so if the odometer rule had a hidden exception the mirroring claim would not survive.
Editorial extensions
If this is right
- For odd $n$, the Collatz step $T(n)=(3n+1)/2$ is a single digit-append operation in rational base $3/2$, while division by $2$ for even $n$ remains a digit-removal in ordinary base $2$; this gives a two-base reading of the Collatz iteration.
- The saturated sequence $\mathrm{sat}(k)$ grows asymptotically like $(3/2)^k$, matching the average growth of $U$.
- The identity $\langle\mathrm{sat}(k-1)\rangle$ being a prefix of $\langle\mathrm{sat}(k)\rangle$ defines an infinite word $\langle\mathrm{sat}(\infty)\rangle$ over $\{1,2\}$; the paper conjectures this word is normal, a statement about the base-$3/2$ numeration analogue of digit randomness.
- The proven equivalence of the Collatz conjecture with the limiting proportion of even terms (Conjectures 2.4 and 2.5) is mirrored by the conjectured limiting proportion of digit $1$ in saturated words, suggesting a transfer between trajectory statistics and word statistics.
Reading between the lines
- A testable extension the paper does not run: compare the finite-$k$ digit-count deviations in $\langle\mathrm{sat}(k)\rangle$ with the statistical bounds that follow from recent almost-all-orbit results for the Collatz map; a match would strengthen the case that Conjecture 4.3 is the right quantitative guess.
- Because the odometer rule is cited rather than proved, an independent implementation of the recursive construction (Proposition 2.2) for a large block of $n$ would be a cheap computational check of the claimed encoding.
- If a digit-level rule for $T(n)$ on even $n$ were found, a question the paper leaves open, the Collatz map would become a finite-state transduction on base-$3/2$ words, making the conjecture amenable to automata-theoretic tools.
Original abstract
The representation of numbers in rational base $p/q$ was introduced in 2008 by Akiyama, Frougny & Sakarovitch, with a special focus on the case $p/q=3/2$. Unnoticed since then, natural questions related to representations in that specific base turn out to intimately involve the Collatz $3x+1$ function. Our purpose in this note is to expose these links and motivate further research into them.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
The representation of numbers in rational base $p/q$ was introduced in 2008 by Akiyama, Frougny & Sakarovitch, with a special focus on the case $p/q=3/2$. Unnoticed since then, natural questions related to representations in that specific base turn out to intimately involve the Collatz $3x+1$ function. Our purpose in this note is to expose these links and motivate further research into them.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The representation of numbers in rational base $p/q$ was introduced in 2008 by Akiyama, Frougny & Sakarovitch, with a special focus on the case $p/q=3/2$. | background 1.00 | 0.03 | 0.04 | asserted 1.00 |
| s1 | Unnoticed since then, natural questions related to representations in that specific base turn out to intimately involve the Collatz $3x+1$ function. | gap 0.49 | 0.51 | 0.45 | asserted 0.99 |
| s2 | Our purpose in this note is to expose these links and motivate further research into them. | contribution 0.56 | 0.39 | 0.19 | asserted 0.73 |
What was examined, and what was not
- arXiv listing abstract v1
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References
Discussion
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