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arxiv: 2605.21098 · v1 · pith:PYBAFB3Jnew · submitted 2026-05-20 · 🧮 math.DS · math.NT· math.PR

A strange continued fraction associated with the Romik map

Yufei Chen , Karma Dajani , Yanyan Hu , Cor Kraaikamp This is my paper

Pith reviewed 2026-05-21 01:53 UTC · model grok-4.3

classification 🧮 math.DS math.NTmath.PR
keywords continued fractionsRomik mapdynamical systemsergodic theorynatural extensionconvergentsPythagorean triplessigma-finite measure
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The pith

The Romik map yields a continued fraction expansion whose partial quotients are restricted to the set {0, ±2} for every x in [0,1].

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper extends analysis of the Romik dynamical system, a modified subtractive Euclidean algorithm originally introduced to generate primitive Pythagorean triples. It constructs a planar natural extension of the system, gives an explicit formula for its sigma-finite invariant measure, and proves ergodicity using standard arguments from infinite ergodic theory. From these dynamical properties the authors derive that, for Lebesgue-almost every real number x, asymptotically half of its ordinary continued-fraction convergents coincide with the convergents produced by the Romik map. They then associate to the same map a continued-fraction expansion in which the only allowed digits are 0, +2 and -2, and record several arithmetic and ergodic properties of this expansion.

Core claim

The central claim is that the Romik map produces a continued fraction in which every x in [0,1] admits an expansion whose partial quotients lie in the three-element set {0, +2, -2}. This expansion is obtained directly from the iterates of the Romik map; its convergents are a subset of the ordinary continued-fraction convergents; and, by ergodicity of the natural extension, the proportion of ordinary convergents that appear as Romik convergents tends to one-half for Lebesgue-almost every x.

What carries the argument

The Romik map itself, whose orbits generate both the restricted-digit continued fraction and the subset of ordinary convergents whose asymptotic density is one-half.

If this is right

  • For Lebesgue-almost every x the Romik convergents form an asymptotic subset of density one-half inside the sequence of ordinary continued-fraction convergents.
  • Every real number in [0,1] possesses at least one expansion whose partial quotients belong exclusively to {0, +2, -2}.
  • Rational numbers and quadratic irrationals possess finite or eventually periodic Romik expansions.
  • The explicit invariant measure on the natural extension permits computation of almost-sure frequencies of digit sequences in the strange expansion.
  • Ergodicity implies that Birkhoff averages of integrable observables along Romik orbits equal the integral against the invariant measure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The three-digit restriction may produce continued-fraction approximations whose denominators grow at a different average rate from the classical golden-ratio bound.
  • The same construction could be applied to other subtractive Euclidean algorithms to obtain further restricted-digit expansions linked to arithmetic objects such as Pythagorean triples.
  • The explicit measure opens the possibility of computing the Hausdorff dimension of the set of x whose Romik expansion avoids a particular digit.
  • Joint dynamics on the natural extension may yield limit theorems for the discrepancy between Romik and ordinary convergents.

Load-bearing premise

The Romik system admits a planar natural extension whose sigma-finite invariant measure can be written explicitly and is ergodic.

What would settle it

Pick a large sample of x drawn uniformly from [0,1], compute the first several hundred Romik convergents and ordinary convergents for each, and check whether the proportion of matches approaches one-half; or verify that a concrete quadratic irrational such as the golden ratio has an infinite expansion using only the digits 0, +2 and -2.

Figures

Figures reproduced from arXiv: 2605.21098 by Cor Kraaikamp, Karma Dajani, Yanyan Hu, Yufei Chen.

Figure 1
Figure 1. Figure 1: The Romik map R [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left: the space of the natural extension [0, 1] × [0, 1]. Right: the image of [0, 1] × [0, 1] under R. Let x ∈ (0, 1) has RCF-expansion x = [0; a1, a2, . . . ], then in Proposition 2.2 we expressed the RCF￾expansion of R(x) in terms of the RCF-expansion of x. Let π2 be the projection of a 2-dimensional vector (x, y) on its second coordinate (i.e., π2(x, y) = y, for all x, y ∈ R), then we define the second … view at source ↗
Figure 3
Figure 3. Figure 3: The set [ 1 n+1 , 1 n ] × [ 1 m+1 , 1 m ] in O, and its image under R. Here n ≥ 3 (ii) Let 1 3 ≤ x ≤ 1 2 . Then from (17) it is immediately clear that R([ 1 3 , 1 2 ] × [0, 1]) = [0, 1] × [ 1 3 , 1 2 ], and again k = 1 = ℓ, and we have the same conclusion as in case (i). (iii) Let 1 2 ≤ x < 1. Then there exist n, m ∈ N, n ≥ 1, and m ≥ 2, such that n n+1 ≤ x < n+1 n+2 , and 1 m+1 ≤ y ≤ 1 m . Since R([ 1 2 ,… view at source ↗
read the original 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 $\sigma$-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 $\sigma$-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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript extends the 2008 Romik dynamical system (a modified subtractive Euclidean algorithm linked to primitive Pythagorean triples) by determining expansions of rationals and quadratic irrationals, constructing a planar natural extension with an explicit σ-finite invariant measure, proving ergodicity via standard arguments in infinite ergodic theory, establishing that Lebesgue-almost every x has asymptotically half its regular continued fraction convergents among the Romik convergents, and introducing a 'strange' continued fraction whose partial quotients for any x in [0,1] lie in the set {0, ±2}, together with various properties of this expansion.

Significance. If the results hold, the work advances understanding of continued fraction expansions through the lens of infinite-measure ergodic theory, providing an explicit natural extension and measure that could support further computations or generalizations. The explicit constructions and the link between Romik convergents and regular continued fraction convergents offer concrete, potentially falsifiable statements about asymptotic densities. The strange continued fraction adds a novel object with a highly restricted digit set, which may prove useful in Diophantine approximation studies.

major comments (1)
  1. [Abstract / natural extension and ergodicity section] Abstract and the section deriving the asymptotic density: the central claim that for Lebesgue-a.e. x asymptotically half the regular continued fraction convergents coincide with Romik convergents is obtained from the planar natural extension and its ergodicity. In infinite ergodic theory, ergodicity alone does not imply the existence of asymptotic densities for the counting functions of visits; the Hopf ratio ergodic theorem (or an equivalent statement on ratios of return times) is required, together with suitable integrability of the relevant functions with respect to the given σ-finite measure. The manuscript states that ergodicity 'follows from standard arguments' but does not appear to verify these ratio conditions or integrability for the convergent-counting functions; this step is load-bearing for the density result and needs explicit justification or a reference to a theorem that,
minor comments (2)
  1. [Section introducing the Romik expansion] The definition of the strange continued fraction map (or algorithm) that produces only digits in {0, ±2} should be stated explicitly, perhaps with a short algorithmic description or recurrence, to make the construction self-contained.
  2. Notation for the Romik map, its natural extension, and the strange expansion should be introduced with a clear table or diagram distinguishing them from the classical Gauss map and regular continued fractions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying this important point regarding the application of infinite ergodic theory. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract / natural extension and ergodicity section] Abstract and the section deriving the asymptotic density: the central claim that for Lebesgue-a.e. x asymptotically half the regular continued fraction convergents coincide with Romik convergents is obtained from the planar natural extension and its ergodicity. In infinite ergodic theory, ergodicity alone does not imply the existence of asymptotic densities for the counting functions of visits; the Hopf ratio ergodic theorem (or an equivalent statement on ratios of return times) is required, together with suitable integrability of the relevant functions with respect to the given σ-finite measure. The manuscript states that ergodicity 'follows from standard arguments' but does not appear to verify these ratio conditions or integrability for the convergent-counting functions; this step is load-bearing for the density result and needs an

    Authors: We agree that ergodicity of the natural extension by itself does not suffice to establish the existence of the asymptotic density for the proportion of coinciding convergents; the Hopf ratio ergodic theorem (or an equivalent result on ratios of return times) must be invoked, and this requires verifying that the relevant counting or indicator functions are integrable with respect to the explicit σ-finite invariant measure. The manuscript indeed invokes 'standard arguments' for ergodicity without spelling out the integrability step for these particular functions. In the revised version we will add a short paragraph (or subsection) in the natural-extension section that explicitly checks the integrability condition for the functions that track visits to the sets corresponding to Romik convergents. We will either give a direct estimate using the explicit form of the measure or cite a suitable theorem from the literature on infinite ergodic theory that guarantees the ratio limit exists and equals the ratio of the integrals. This will make the derivation of the density result fully rigorous. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via explicit measure and external 2008 reference

full rationale

The paper cites the 2008 Romik construction (different author) only as background for the original system and then independently constructs a planar natural extension, supplies an explicit σ-finite invariant measure, and establishes ergodicity via standard infinite-measure arguments. The asymptotic-density claim for Romik versus regular continued-fraction convergents is presented as a consequence of this new extension and measure. No equation or definition in the provided text reduces a claimed result to a fitted parameter, a self-referential definition, or a load-bearing self-citation chain; the cited 2008 result is externally falsifiable and does not contain the target density statement. The derivation therefore remains independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on the prior definition of the Romik map and on standard theorems from ergodic theory for maps preserving sigma-finite infinite measures; no new free parameters or postulated entities are introduced.

axioms (1)
  • standard math Standard results from ergodic theory apply to the natural extension of the Romik map on the unit interval.
    Invoked when the abstract states that the natural extension is ergodic and possesses an explicit sigma-finite invariant measure.

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Reference graph

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