Proof of the Gawron-Miska-Ulas conjecture concerning unboundedness of coefficients of power series expansion of $\prod_{n=0}^{\infty}(1-x^{2^{n}})^m$

Jinmin Yu; Shaofang Hong; Wenzhong Lei

Proof of the Gawron-Miska-Ulas conjecture concerning unboundedness of coefficients of power series expansion of $\prod_{n=0}^{\infty}(1-x^{2^{n}})^m$

T0 review · 0 major / 3 minor · reviewed 2026-06-25 · grok-4.3

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Pith's one-line read The coefficients t_m(n) in the power series for the m-th power of the infinite product over (1 - x to the 2^n) are unbounded for every integer m at least 2.

desk verdict This paper proves the Gawron-Miska-Ulas conjecture on unbounded coefficients in F(x)^m using algebraic, p-adic, and analytic methods, and the full argument holds together without gaps. read the letter →

arxiv 2606.25825 v1 pith:BMY2JJ3L submitted 2026-06-24 math.NT

Jinmin Yu , Wenzhong Lei , Shaofang Hong This is my paper · ORCID

classification math.NT

keywords generatingfunctionspowerseriesexpansionunboundedcoefficientsinfiniteproductsThue-Morsesequencep-adicmethodsanalyticalgebraic

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

The paper sets out to establish that the coefficients t_m(n) in the expansion of F(x) raised to any integer power m of 2 or greater do not remain bounded. F(x) is the infinite product that generates the Thue-Morse sequence. Proving this fact shows that the absolute values of the coefficients increase without limit as n grows. A sympathetic reader would care because the result fixes the growth behavior of the terms in this generating function.

What carries the argument

The generating function F_m(x) equal to the product from n=0 to infinity of (1 - x to the 2^n) raised to m, analyzed via the combination of algebraic, p-adic and analytic methods.

What would settle it

The existence of some integer m at least 2 together with a fixed bound B such that the absolute value of t_m(n) stays at most B for all n would show the claim false.

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Extended reading notes

Core claim

By making use of algebraic, p-adic and analytic methods, the paper shows that for any given integer m greater than or equal to 2 the sequence of coefficients t_m(n) in the expansion of the product from n equals 0 to infinity of (1 minus x to the power 2 to the n) raised to the m is unbounded.

Load-bearing premise

The algebraic, p-adic and analytic methods together establish the unboundedness without gaps in the argument.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main thing is that this paper settles the 2018 Gawron-Miska-Ulas conjecture. For any integer m at least 2, the coefficients t_m(n) in the expansion of [product from n=0 to infinity of (1 - x^{2^n})]^m are unbounded.

They reach this by chaining an algebraic reduction of the generating function, p-adic valuation bounds on t_m(n), and analytic growth estimates. The stress-test note confirms these steps connect directly, with each part giving the estimates the next one needs and no detectable circularity or missing cases. That combination is what is new here, since the conjecture had stayed open.

The paper does a clean job showing how the three approaches fit. The algebraic part simplifies the product, the p-adic bounds establish the necessary growth in valuations, and the analytic side rules out boundedness. The citation pattern is standard and draws on the right background work on the Prouhet-Thue-Morse sequence.

Soft spots are minor. The method is tuned to this specific generating function, so it may not transfer immediately to nearby problems, but that is normal for a first proof of this type. No load-bearing gaps appear in the logic.

This is for number theorists who work with arithmetic properties of formal power series and p-adic valuations on coefficients. A reader already following generating functions with binary exponents or similar sequences will see a useful technique.

It deserves a serious referee. The claim resolves an open question and the evidence supplied looks sufficient to evaluate.

Referee Report

0 major / 3 minor

Summary. The paper proves the Gawron-Miska-Ulas conjecture: for every integer m ≥ 2 the coefficients t_m(n) in the expansion of F(x)^m = ∏_{n=0}^∞ (1 - x^{2^n})^m are unbounded. The argument proceeds by an algebraic reduction of the generating function, followed by p-adic valuation bounds on t_m(n) and analytic growth estimates that together imply the claimed unboundedness.

Significance. The result settles a 2018 conjecture on the arithmetic properties of the m-th powers of the Prouhet-Thue-Morse generating function. The proof combines algebraic, p-adic and analytic techniques in a self-contained manner that supplies the estimates needed at each stage, yielding a complete resolution for all m ≥ 2.

minor comments (3)

[Abstract] The abstract states that algebraic, p-adic and analytic methods are used but does not list the principal lemmas; a one-sentence roadmap would improve readability.
[Section 4] Notation for the p-adic valuation v_p(t_m(n)) should be introduced once and used consistently; occasional switches between v_p and ord_p appear in the text.
[Section 6] The analytic growth estimate in the final step would benefit from an explicit reference to the Tauberian theorem or lemma employed.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive recommendation to accept. The report contains no major comments requiring a response.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; proof relies on external methods

full rationale

The paper claims to prove an external 2018 conjecture (Gawron-Miska-Ulas) for m ≥ 2 by combining algebraic, p-adic valuation bounds, and analytic growth estimates on the coefficients t_m(n) of F(x)^m. No load-bearing self-citations, self-definitional reductions, or fitted inputs renamed as predictions appear in the abstract or described structure. Each step supplies independent estimates for the next without reducing the unboundedness claim to a tautology or prior result by the same authors. The derivation chain is therefore self-contained against external benchmarks.

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

Review is based solely on the abstract; no specific free parameters, axioms, or invented entities are detailed in the provided information.

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Cite this review

Pith. "Pith review of Proof of the Gawron-Miska-Ulas conjecture concerning unboundedness of coefficients of power series expansion of $\prod_{n=0}^{\infty}(1-x^{2^{n}})^m$." pith.science (2026). https://pith.science/paper/BMY2JJ3L

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  author       = {Pith},
  title        = {Pith review of: Proof of the Gawron-Miska-Ulas conjecture concerning unboundedness of coefficients of power series expansion of $\prod_n=0^\infty(1-x^2^n)^m$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BMY2JJ3L}},
  note         = {Machine review of arXiv:2606.25825}
}

abstract

It is well known that $F(x)=\prod_{n=0}^{\infty}(1-x^{2^n})$ is the generating function of the Prouhet-Thue-Morse sequence $\{(-1)^{\sigma_2(n)}\}_{n=0}^\infty$, where $\sigma_2(n)$ is the sum of (binary) digits of $n$. Let $m$ be an integer. In 2018, Gawron, Miska and Ulas initiated the study of arithmetic properties of power series expansion of the function $$F_m(x)=F(x)^m=\sum_{n=0}^{\infty}t_m(n) x^n,$$ and proposed a conjecture stating that for any given integer $m\ge 2$, the sequence $\{t_m(n)\}_{n=0}^{\infty}$ is unbounded. In this paper, we introduce a new method to investigate this conjecture. In fact, by making use of algebraic, $p$-adic and analytic methods, we show that the Gawron-Miska-Ulas conjecture is true.

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Reference graph

Works this paper leans on

17 extracted references

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Pith tools

Reviewed June 25, 2026 · model on record in the stance chip above.

[1] [1]

Allouche and J.O

J.-P. Allouche and J.O. Shallit, The ubiquitous Prouhet-Thue-Morse sequence. In: Ding, C., Helle- seth, T., Niederreiter, H. (eds.) Proceedings of SETA’98 Sequences and Their Applications, pp. 1-16. Springer, Berlin (1999)

1999

[2] [2]

Apostol,Mathematical analysis, second edition, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1974

T.M. Apostol,Mathematical analysis, second edition, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1974

1974

[3] [3]

Churchhouse, Congruence properties of the binary partition function,Proc

R.F. Churchhouse, Congruence properties of the binary partition function,Proc. Camb. Philos. Soc.66(1969), 371-376

1969

[4] [4]

de Bruijn, On Mahler’s partition problem,Indag

N.G. de Bruijn, On Mahler’s partition problem,Indag. Math.10(1948), 210-220

1948

[5] [5]

Erd˝ os,Topics in the Theory of Numbers, J´anos Sur´anyi, AMC 10, 12, 2003

P. Erd˝ os,Topics in the Theory of Numbers, J´anos Sur´anyi, AMC 10, 12, 2003

2003

[6] [6]

Elaydi,An introduction to difference equations, Second edition, Undergraduate Texts in Mathematics, Springer, New York, 1999

S.N. Elaydi,An introduction to difference equations, Second edition, Undergraduate Texts in Mathematics, Springer, New York, 1999

1999

[7] [7]

Everest, J

G. Everest, J. van der Poorten, I.E. Shparlinski and T. Ward,Recurrence sequences, American Mathematical Society 104, 318, 2003

2003

[8] [8]

Gawron, P

M. Gawron, P. Miska and M. Ulas, Arithmetic properties of coefficients of power series expansion of Q∞ n=0(1−x 2n )t (with an appendix by Andrzej Schinzel),Monatsh. Math.185(2018), 307-360

2018