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REVIEW 4 major objections 6 minor 1 cited by

Exercises in Iterational Asymptotics

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For three nonlinear recurrences, including $x_k=x_{k-1}(1-\sqrt{x_{k-1}})$ and $x_k=x_{k-1}+1/x_{k-1}^{q-1}$, explicit asymptotic expansions are derived and the new constants in them are evaluated to high precision.

desk verdict A useful collection of new explicit expansions and constants for nonlinear recurrences, undercut by heavy reliance on an unproved self-cited template; deserve a careful referee, not a desk reject. read the letter →

arxiv 2411.16062 v3 pith:BAOP4TOG submitted 2024-11-25 math.NT cs.DMmath.CAmath.CO

Steven Finch This is my paper · ORCID
classification math.NTcs.DMmath.CAmath.CO MSC 39A1040A0565Q20
keywords nonlinearrecurrencesasymptoticexpansioniterateslogisticmapslowdecaynumericalconstantsreciprocitypowerseriesmatching
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper presents worked asymptotic analyses of three families of one-dimensional nonlinear recurrences. For the map $x_k=x_{k-1}(1-\sqrt{x_{k-1}})$, it derives a complete expansion in powers of $1/k$ and $\log k$ through $1/k^6$, including the new numerical constant $C=1.98803983644549695008812308629512\ldots$ for the midpoint initial value, and analogous constants for a starting value that minimizes $C$. For the slow-growth recurrence $x_k=x_{k-1}+1/x_{k-1}^{q-1}$ with $q>1$, it derives expansions in powers of $k^{-1/q}$ and $\log k$ and reports constants $c(2)=0.8615711875687117305317813\ldots$, $c(3)=1.3784186157718345713984647\ldots$, and $c(3/2)=0.8010888849039666437110775\ldots$, with the last computed via a reciprocity relation stated in the addendum. A first exercise determines the geometric decay constant for $p$-scaled logistic maps $x_k=p\,x_{k-1}(1\pm x_{k-1})$. The paper presents these as new problems and solutions; if correct, they give explicit closed-form asymptotics where only numerical iteration was previously available.

What carries the argument

The argument rests on an asymptotic template for iterations $f(x)=x(1-a x^\tau)$: the paper assumes the form $x_k \sim (\lambda/k)^{1/\tau}\{1+\sum_{m=1}^6 P_m(-(1/\tau)(b_1\ln(k)+C))/k^m\}$, where $\tau$, $\lambda$, and the sequences $a_m$, $b_j$, $c_i$ determine the polynomials $P_m$; for $f(x)=x(1-\sqrt{x})$ this is $\tau=1/2$, $\lambda=2$, and it produces the displayed expansion through $1/k^6$. For the $q$-recurrence, the transformation $y_k=x_k^q$ reduces to $y_k=y_{k-1}(1+1/y_{k-1})^q$, whose asymptotics follow from cited theorems for iterates of functions with $\phi(0)=1$ and $\phi'(0)\neq0$; a brute-force matching-coefficient method, comparing series for $x_{k+1}$ and $x_k+x_k^{1-q}$, extends the expansion to higher order. The addendum's reciprocity relation---that a limit involving iterates of $\xi_k=\xi_{k-1}/(1+\xi_{k-1}^{3/2})$ equals $2\Lambda/3$ with $\Lambda=c(3/2)$---brings the hard $q=3/2$ case back into the range of the earlier convergent-iteration method.

What would settle it

Take $x_0=1/2$, iterate $x_k=x_{k-1}(1-\sqrt{x_{k-1}})$ in high-precision arithmetic up to $k=10^6$, and compare $x_k$ to the six-term expansion with $C=1.98803983644549695008812308629512$. If the difference after subtracting all displayed terms through order $1/k^6$ does not decay like $o(k^{-6})$ (or at least like $O(k^{-7})$), then the template or the constant is wrong. For $q=2$, iterate $x_k=x_{k-1}+1/x_{k-1}$ from $x_0=1$ and test that $2^{1/2}x_k - 2k^{1/2} - \frac{1}{4}\frac{\ln(k)}{k^{1/2}} - c(2)\frac{1}{k^{1/2}}$ converges to $0$ at the predicted rate.

Watch

Extended reading notes

Core claim

The central discovery is that iterates of $f(x)=x(1-\sqrt{x})$ have the asymptotic form $x_k \sim 4/k^2 - 12\ln(k)/k^3 - 8C/k^3 + \cdots$, with $C=1.98803983644549695008812308629512\ldots$ when $x_0=1/2$, and that the full expansion through order $1/k^6$ is given by the displayed polynomial template with $P_2$ through $P_6$. The paper likewise establishes that for $x_k=x_{k-1}+x_{k-1}^{1-q}$, substituting $y_k=x_k^q$ yields $y_k \sim q k + \frac{q-1}{2}\ln(k) + C + \cdots$, and it reports $c(2)=0.8615711875687117305317813\ldots$, $c(3)=1.3784186157718345713984647\ldots$, and $c(3/2)=0.8010888849039666437110775\ldots$ after invoking a reciprocal limit in the addendum. All of these are stated as new as far as is known, and are meant to quantify the long-term trend of the respective recurrences exactly.

Load-bearing premise

The expansions all depend on asymptotic formulas imported from earlier work being valid for these particular recurrences; if any recurrence falls outside those formulas' hypotheses, the displayed constants shift.

Editorial extensions

If this is right

  • For $f(x)=x(1-\sqrt{x})$, the expansion through $1/k^6$ lets one compute $x_k$ accurately for large $k$ from the constant $C$ alone, without iterating the map millions of times.
  • For the $q$-recurrence, the displayed formulas give explicit closed-form asymptotics for every $q>1$ at three benchmark values, with $c(2)$, $c(3)$, and $c(3/2)$ as numerical anchors.
  • The addendum shows the constants for divergent iterations where $x_k$ grows like $k^{1/q}$ and convergent iterations where $x_k$ decays are the same constants under a reciprocal transformation, so computing one gives the other.
  • For $p$-scaled logistic maps with $p<1$, the product formula $C=x_0\prod_{j=0}^{\infty}(1-x_j)$ gives a direct way to estimate the geometric decay rate from finitely many iterations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The polynomial template used for $\tau=1/2$ likely applies to other maps tangent to the identity at $0$, such as $u_k=u_{k-1}(1-u_{k-1}^2/2)$, $v_k=v_{k-1}\cos(v_{k-1})$, and $w_k=w_{k-1}\exp(-w_{k-1}^2/2)$, which the paper explicitly leaves open; deriving their $\tau$ and $\lambda$ would give comparable expansions.
  • The reciprocity in the addendum suggests a general two-parameter duality between iterates of $x/(1+x^r)$ and $x+x^{1-q}$; if the duality holds more broadly, the hardest regimes (for example irrational $q$) could be handled by the convergent-iteration method rather than the brute-force matching one.
  • The brute-force matching-coefficient method appears to be a general "shift and compare" tool: because it uses only the recurrence and the leading asymptotics, it could extend many slow recurrences to arbitrary order without additional analytic theorems.
  • The same reciprocity could be pushed further: expanding the companion iterations $\eta_k$ and $\zeta_k$ to higher order should reproduce $c(3)$ and $c(2)$ from a convergent-iteration calculation, making the brute-force step unnecessary at those parameters.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper presents three worked exercises in iterational asymptotics. Section 1 proves existence of the convergence-rate constant C for the recurrences x_k = p x_{k-1}(1 - x_{k-1}) and x_k = p x_{k-1}(1 + x_{k-1}), and tabulates numerical estimates for several p. Section 2 derives a six-term asymptotic expansion for x_k = x_{k-1}(1 - sqrt(x_{k-1})) by applying a template from the author's preprint [3], and reports the constant C = 1.98803983644549695008812308629512... for x0 = 1/2 and a second value for x0 = 4/9. Section 3 derives expansions for x_k = x_{k-1} + x_{k-1}^{1-q} using theorems of Popa [17,18], reports constants c(2), c(3), and c(3/2), and corrects a coefficient in [18]. The Addendum completes the q = 3/2 case via an asserted reciprocity transformation.

Significance. If the imported templates and theorems are valid, the paper provides new explicit asymptotic expansions, new high-precision constants, and a correction of a published coefficient, which would be useful contributions. The existence arguments in Section 1 are complete and self-contained, and the algebra after the imports is coherent. However, the central Section 2 expansion rests entirely on an unpublished preprint by the same author, and the Section 3 results rest on imported theorems with a stated but unproved correction. No error bounds or independent numerical cross-checks are supplied for the many-digit constants. Thus the significance is real but conditional on the validity of the external machinery, which the manuscript does not make independently verifiable.

major comments (4)
  1. [Section 2, template from [3]] The line 'formulaic knowledge of sections 1, 2, 3 of [3] is assumed' is not a substitute for stating the hypotheses of the template. The paper gives no theorem, proof, or precise statement of the class of functions f for which the template applies, and [3] is an unpublished preprint by the same author. Because the displayed polynomials P_2 through P_6, the coefficients {b_j}, {a_0j}, {c_i}, and the resulting six-term expansion all follow from this template, and because C = 1.98803983644549695008812308629512... is estimated using that same expansion, the central numerical claim is not independently verifiable from the manuscript. Please either include the relevant theorem with hypotheses and proof (or a precise published reference), or provide an independent numerical cross-check that validates the first few coefficients.
  2. [Section 5 (Addendum), reciprocity transformation] The reciprocity transformation used to complete the q = 3/2 case is asserted without proof. The Addendum states 'A certain reciprocity has been found' and then gives the limiting relation for ξ_k from which c(3/2) = 0.8010888849039666437110775... follows. Since this is the only route to the reported value, the transformation and the limiting relation need a derivation or a reference containing a proof. As written, the value is an unexplained additional axiom of the paper.
  3. [Section 3, correction to Popa's δ] The note 'beware: the lead coefficient 1/2 of δ in [18] should be 1/4' is a correction to an imported theorem, but the corrected formula is used without re-derivation or numerical test. Because the expansions in this section are only as reliable as the imported theorems, this correction should be justified—ideally by a short derivation of δ or by checking the resulting expansion against a numerical example—rather than stated as a parenthetical warning.
  4. [Section 3, constants precision] The constants c(2), c(3), and c(3/2) are quoted to 25 or more decimal places, but the paper provides no error analysis or convergence test for the 'brute-force matching-coefficient method' that produced them. The reader cannot tell how many digits are reliable. Please report an error estimate or a comparison of successive truncations of the expansions for each constant.
minor comments (6)
  1. [Abstract] The phrase 'a s far as is known' contains a spacing error; it should read 'as far as is known'.
  2. [Section 1] The text 'More ge nerally' has a stray space; it should read 'More generally'.
  3. [Section 2] The parenthetical remark 'Assu ming x0 = 4/9' contains a spacing error; it should read 'Assuming x0 = 4/9'.
  4. [Section 3] The phrase 'blead coefficient' is a typo for 'lead coefficient'.
  5. [Tables 1 and 2] The captions state that 'no closed-form expressions are known'; this is an unsupported empirical claim and should be softened, for example to 'none are given here'.
  6. [Section 3, brute-force method] The sentence 'The elegant technique from [13, 14], useful in Section 2, does not apply here' is clear, but the subsequent description of the brute-force method would benefit from a short example showing how the series for x_{k+1} is compared with the series for x_k + x_k^{-(q-1)}.

Circularity Check

1 steps flagged · score 4.0 of 10

Section 2's expansion and polynomial families are imported wholesale from the author's own unpublished [3], and the headline constant C is estimated from that same self-cited expansion; the derivation is therefore heavily self-referential, though not a definitional reduction.

  1. self citation load bearing [Section 2 (Deuxième exercice), paragraphs 2–3: 'To conserve space...' through 'using the preceding expansion.']
    "To conserve space, formulaic knowledge of sections 1, 2, 3 of [3] is assumed. ... The remarkable formula connecting Pm and asymptotics of xk = f (xk−1) is xk ∼ (λ/k)^{1/τ} { 1 + 6∑ m=1 Pm( − 1/τ [b1 ln(k) + C] ) 1/km } ... Assuming x0 = 1/2 (the midpoint), the constant C is estimated to be C = 1.98803983644549695008812308629512... by a simple numerical method [2] using the preceding expansion."

    The b_j, a_0j, c_i arrays and the polynomials P_2–P_6 that determine the full six-term expansion are not derived in this paper; they are asserted to follow from 'sections 1, 2, 3 of [3]'. Reference [3] is the same author's own arXiv preprint, and the paper states no hypotheses, proof, or independent numerical cross-check for that template. The quoted expansion is then used to estimate the headline constant C, so the central Section 2 result rests on a self-citation that is itself unverified in the present text. This is load-bearing self-citation: the derivation chain for the expansion and for the numerical value of C passes through [3] without any external grounding supplied here.

full rationale

The paper is not a circular derivation in the strict sense: the asymptotic expansions are not defined in terms of the constants being estimated, and the constants C are obtained numerically from the recurrences using the expansions, which is a standard coefficient-matching procedure. Section 3 relies on Popa's external theorems [17,18], and the paper even flags and corrects an error in [18] ('the lead coefficient 1/2 of δ in [18] should be 1/4'), which shows the imported results are being checked rather than merely assumed. The main circularity concern is Section 2, where the central expansion is imported wholesale from the author's own unpublished preprint [3] with the phrase 'formulaic knowledge ... is assumed'; no proof, assumptions, or external verification is given, and the same expansion is used to estimate the paper's headline constant. This raises the score to 4 (some self-citation; the central claim still has independent content in the numerically evaluated constants and the explicit application to new f), but it does not justify a higher score such as 6 or 8, because no equation is shown to reduce to another by construction and no fitted parameter is relabeled as a prediction.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central expansion formulas and the numerical constants hinge on the validity of imported asymptotic templates, particularly the author's own [3], and on numerical methods that are not fully described. The C and c(q) values are not derived in closed form and must be taken on the author's numerical computation.

free parameters (4)
  • C(x0), Section 2 integration constant = 1.98803983644549695008812308629512... for x0=1/2; 1.96846882098495471088450855794395... for x0=4/9
    Appears as an undetermined constant in the asymptotic expansion; no closed form is given, and it is estimated numerically using the expansion itself.
  • c(2) = C(2)/2 = 0.8615711875687117305317813...
    Scaled limit constant for the q=2 recurrence, obtained by brute-force matching-coefficient method.
  • c(3) = C(3)/3 = 1.3784186157718345713984647...
    Scaled limit constant for the q=3 recurrence, obtained by brute-force matching-coefficient method.
  • c(3/2) = Lambda = 0.8010888849039666437110775...
    Scaled limit constant for the q=3/2 recurrence, completed via reciprocity in the Addendum with details deferred to [20].
assumptions (4)
  • domain assumption The general asymptotic expansion template involving the polynomials P_m, stated in [3], is valid and applies to x_k = x_{k-1}(1 - sqrt(x_{k-1})).
    Section 2: 'formulaic knowledge of sections 1, 2, 3 of [3] is assumed'; the formula is not proved in this paper.
  • domain assumption Theorems 5 and 6 and Proposition 7 of Popa [17,18] are correct and applicable to y_k = y_{k-1}(1 + 1/y_{k-1})^q.
    Section 3 imports the y_k expansion with 'By Theorem 6 of [17] and Theorem 5 of [18]', including a correction to a coefficient in [18].
  • ad hoc to paper The brute-force matching-coefficient method yields the displayed additional expansion terms and c(q) values to the stated precision.
    Section 3 describes the method only by reference to [2,4,15,19]; no error analysis or convergence proof is given.
  • ad hoc to paper The reciprocity transformation maps the q=3/2 case to the convergent iteration xi_k and yields c(3/2) = Lambda.
    Addendum states the reciprocity and value, with details deferred to the author's [20].

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Cite this review

Pith. "Pith review of Exercises in Iterational Asymptotics." pith.science (2026). https://pith.science/paper/BAOP4TOG

@misc{pith2026241116062,
  author       = {Pith},
  title        = {Pith review of: Exercises in Iterational Asymptotics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BAOP4TOG}},
  note         = {Machine review of arXiv:2411.16062}
}
read the original abstract

The problems and solutions contained here, all associated with nonlinear recurrences and long-term trends, are new (as far as is known).

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Popa's "Recurrent Sequences" and Reciprocity

    math.CA 2024-12 conditional novelty 5.0 of 10

    For a family of nonlinear recurrences, the paper evaluates the asymptotic constant C to 25 digits using a reciprocal transformation and the Mavecha-Laohakosol algorithm.

Reference graph

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