arXiv:2605.21098v2

A strange continued fraction associated with the Romik map

Yufei Chen, Karma Dajani, Yanyan Hu, Cor Kraaikamp

math.DSmath.NTmath.PR11J7028D05

Paper metadata

Original submission: May 20, 2026
Latest revision: July 1, 2026 · version 2
Claimed MSC: 11J70, 28D05
arXiv categories: math.DS, math.NT, math.PR
Paper license: arXiv non-exclusive distribution license ↗
Comments: 29 pages, 3 figures

Abstract

In 2008, Dan Romik studied in this journal Primitive Pythagorean Triples, or PPTs. In order to do so, he introduced a modified slow (subtractive) Euclidean algorithm, and showed that the underlying dynamical system of this Euclidean algorithm (the ``Romik system''), is ergodic and has a $σ$ -finite, infinite measure, of which is explicitly given. In this paper, the Romik system is further studied. Various basic properties are determined, such as the expansion of rational numbers and quadratic irrationals. Also (a version of) the planar natural extension of the Romik system is obtained, and the $σ$ -finite, invariant measure is explicitly given, and it is shown that it is ergodic. Furthermore, for Lebesgue almost every $x$ asymptotically half of the regular continued fraction (RCF) convergents of $x$ are among the Romik convergents. We also show that related to the Romik map a ``strange'' continued fraction can be given. ``Strange,'' as the set of possible partial quotients (i.e., digits) for any $x\in [0,1]$ in this expansion is $\{ 0, \pm 2\}$ . Various properties of this ``Romik expansion'' are given.

AI-generated audit

Audit summary

Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 15, 2026

01Statements3 reported findingsCorrect

The central statements checked are correct. The rational and quadratic-orbit results, the natural extension and invariant measure, the formal Romik expansion and convergence results, the conversion algorithm, and the almost-everywhere one-half deletion law follow from the supplied arguments. The literal notation defects listed in Part 2 have mechanically determined repairs and do not change these conclusions.

Propositions 2.1–2.5Correct

Arithmetic and dynamical foundations of the Romik map

Pages 5–11 · Propositions 2.1–2.5 · arXiv:2605.21098v2

For rational inputs, the branch formulas strictly reduce an appropriate denominator until the orbit reaches an endpoint. The regular-continued-fraction branch identities then imply eventual periodicity for quadratic irrationals. For the planar extension, the inverse branches, Jacobians, and density calculation give the invariant density $(x+y-2xy)^{-2}$ and its marginal $1/(x(1-x))$ ; the branch coding supplies the asserted natural-extension and ergodicity conclusions.

Full paper, version 2 ↗

Theorem 3.1 and Proposition 4.3Correct

Formal expansion, convergents, and approximation identities

Pages 17–24 · Theorem 3.1, Lemma 4.1, Proposition 4.3, and Lemma 4.4 · arXiv:2605.21098v2

The branch-by-branch induction produces the stated digits and signs, and the matrix recurrence gives the convergents and determinant identity. Substitution of the tail $R^k(x)$ yields the two exact approximation formulas after the absolute-value typo below is corrected. The analysis of repeated-convergent blocks proves that, for irrational $x$ , the denominators eventually exceed every prescribed bound.

Full paper, version 2 ↗

Algorithm 1 and Theorem 5.2Correct

Conversion from regular continued fractions and the metric deletion law

Pages 25–28 · Lemma 5.1, Algorithm 1, and Theorem 5.2 · arXiv:2605.21098v2

The insertion and singularization identities convert each regular partial quotient into the allowed Romik blocks and preserve the represented number. After the mechanically determined index corrections listed below, the algorithm covers the three possible first non- $2$ cases. Hopf's ratio theorem applies to the two indicator functions; the invariant-density integrals are $\log 2$ and $2\log 2$ , giving the asserted ratio $1/2$ .

Full paper, version 2 ↗

02Proofs5 reported findingsCorrect

The proofs of the central results are correct and complete after immediate-consequence closure. The findings below are genuine literal notation defects whose intended corrections are fixed by the adjacent matrix calculations or by the algorithm's own case definitions. They require textual correction but do not lower the proof status.

Central proof chainCorrect and complete

Branch identities, matrix recurrences, and ergodic ratio argument

Pages 5–28 · Sections 2–5 · arXiv:2605.21098v2

The denominator descent, finite-state periodicity argument, inverse-branch computation, continued-fraction matrix products, convergence analysis, and ratio-ergodic calculation supply every substantive step used by the main statements. The short natural-extension argument is recoverable from the displayed inverse branches and the generating branch coding; no additional estimate, compactness argument, or change of quantifiers is needed.

Full paper, version 2 ↗

Proposition 4.3(ii)Typo

The approximation formula is missing absolute-value bars

Pages 22–23 · Equation (38) and its proof · arXiv:2605.21098v2

Equation (38) prints $x-p_n/q_n$ equal to a positive expression. Replace the left-hand side by $\lvert x-p_n/q_n\rvert$ . The determinant identity in Lemma 4.1 fixes the omitted sign, and the proof immediately below computes the magnitude. For example, the displayed sample has $x< p_1/q_1$ , so the unsigned printed equality cannot hold. The corrected identity is exactly the estimate used later.

Full paper, version 2 ↗

Lemma 5.1Typo

The inserted-convergent indices are duplicated

Page 25 · Statement and matrix calculation in Lemma 5.1 · arXiv:2605.21098v2

The first inserted convergent is printed with denominator $2s_{n-1}+s_{n-1}$ ; replace the second $s_{n-1}$ by $s_{n-2}$ . In the subsequent product, replace the duplicated terms such as $2r_{n-2}+r_{n-2}$ by $2r_{n-1}+r_{n-2}$ , and make the analogous denominator correction. Direct multiplication of the displayed matrices uniquely gives these indices, and the final matrix already uses the correct recurrence.

Full paper, version 2 ↗

Algorithm 1Typo

Five symbols and indices contradict the surrounding case definitions

Pages 26–27 · Steps I–II and Equations (42)–(44) · arXiv:2605.21098v2

Step I defines $n$ as the first index with $a_n\ne2$ , so its following infimum and empty-set condition must also use $a_n\ne2$ , not $a_n=2$ . In the odd case, the exponent in Equation (42) must be $(a_n-1)/2$ , matching the stated number of insertions. In case (iii), the number of insertions is $a_{n+1}-1$ , not $a_n-1$ . In Step II, the terminal case is $d_n=2$ for every $n$ , not $d_n\ne2$ for every $n$ . Finally, the continued fraction produced in case (iii) is Equation (44), not Equation (43). Each correction is forced by the immediately adjacent transformation and leaves the represented continued fraction unchanged.

Full paper, version 2 ↗

Lemma 4.1 and its setupTypo

The matrix group and greatest-common-divisor notation have the wrong signs

Pages 19–20 · Paragraph before Remark 4.1 and Lemma 4.1(ii) · arXiv:2605.21098v2

Matrices of determinant $\pm1$ belong to $\mathrm{GL}_2(\mathbb Z)$ , not necessarily $\mathrm{SL}_2(\mathbb Z)$ , so replace the latter group name in this setup by the former. Also replace $\gcd(p_n,q_n)=\pm1$ by $\gcd(p_n,q_n)=1$ . The displayed determinant formula uniquely determines both corrections, and every subsequent Möbius and coprimality argument uses only determinant $\pm1$ .

Full paper, version 2 ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

Detailed audit reportFull reasoning, manuscript locations, and references.

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Paper: arXiv:2605.21098v2
Authors listed: Yufei Chen, Karma Dajani, Yanyan Hu, Cor Kraaikamp
Audit date: August 15, 2026

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You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. 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If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. Do not report a gap, unsupported statement, minor formal correction, or recommendation to add an explanatory sentence, and do not mention the suppressed item in the report. This suppression does not apply to a genuine typographical error as defined below, because a typo is a literal notation defect that still requires a textual correction. For example, if

P(delta)

is proved for every sufficiently small

delta>0

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P

is immediately monotone in the required direction, then the claim that

P(delta)

holds for every

delta>0

requires no correction; the omitted monotonicity step is exposition rather than an audit finding. 6. Immediate-consequence closure permits reconstruction of an omitted inference; it does not permit silent alteration of the printed domain, hypotheses, or quantifiers of an intermediate statement later invoked outside its stated scope. If formal validity requires changing the text of that statement or its stated range, report the appropriate finding even when the repair is easy. Thus a result stated only for

t>0

and subsequently applied at

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A strange continued fraction associated with the Romik map

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Arithmetic and dynamical foundations of the Romik map

Formal expansion, convergents, and approximation identities

Conversion from regular continued fractions and the metric deletion law

Branch identities, matrix recurrences, and ergodic ratio argument

The approximation formula is missing absolute-value bars

The inserted-convergent indices are duplicated

Five symbols and indices contradict the surrounding case definitions

The matrix group and greatest-common-divisor notation have the wrong signs

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