Unboundedness of the Coefficients of Higher Powers of a Unimodular Power Series

Zhao Shen

arxiv: 2606.28411 · v1 · pith:55W732LUnew · submitted 2026-06-25 · 🧮 math.CO

Unboundedness of the Coefficients of Higher Powers of a Unimodular Power Series

Zhao Shen This is my paper

Pith reviewed 2026-06-30 01:33 UTC · model grok-4.3

classification 🧮 math.CO

keywords unimodular power seriescoefficient unboundednessformal power seriesParseval identityJensen inequalityunit circle coefficients

0 comments

The pith

For any power series with all coefficients of modulus one, the coefficients of its m-th power are unbounded when m is at least 2.

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

The paper shows that if a formal power series R(z) has every coefficient r_n satisfying |r_n| = 1, then for each integer m at least 2 the coefficients in the expansion of R(z)^m form an unbounded sequence. The argument applies Parseval's identity to relate the sum of squared coefficients to an integral over the unit circle and then invokes Jensen's inequality on a convex function of the logarithm. A reader would care because the result directly confirms Conjecture 3.9 from earlier work on unimodular series and rules out the possibility that all higher powers could simultaneously have bounded coefficients.

Core claim

We show that for each integer m≥2, the coefficient sequence of R(z)^m is unbounded. The proof combines Parseval's identity with Jensen's inequality. As a consequence, Conjecture 3.9 of Gawron, Miska, and Ulas is confirmed.

What carries the argument

The direct application of Parseval's identity together with Jensen's inequality to the formal power series R(z) whose coefficients lie on the unit circle.

Load-bearing premise

That Parseval's identity and Jensen's inequality can be applied directly to the formal power series R(z) with |r_n|=1 for all n to conclude unboundedness of the coefficients of R(z)^m.

What would settle it

An explicit sequence r_n with |r_n|=1 for all n such that the coefficients of R(z)^m remain bounded for some fixed m≥2.

read the original abstract

Let $R(z)=\sum_{n=0}^{\infty} r_n z^n$ be a power series with $|r_n|=1$ for every $n\ge 0$. We show that for each integer $m\ge 2$, the coefficient sequence of $R(z)^m$ is unbounded. The proof combines Parseval's identity with Jensen's inequality. As a consequence, Conjecture~3.9 of Gawron, Miska, and Ulas \cite{gmu} is confirmed.

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.

Desk Editor's Note private letter to a colleague

The paper proves the Gawron-Miska-Ulas conjecture on unbounded coefficients in powers of unimodular series via Parseval plus Jensen, but the direct application to arbitrary phases needs explicit justification for non-convergent cases.

read the letter

The main result is a proof that if |r_n|=1 for all n, then the coefficients of R(z)^m grow unbounded for every m at least 2. This settles Conjecture 3.9 from Gawron-Miska-Ulas. The argument is short and rests on two standard tools rather than new machinery.

The paper does one thing cleanly: it shows how those two inequalities together force the conclusion once the setup is in place. No extra parameters or ad-hoc constructions appear.

The soft spot is exactly the one flagged in the stress test. Parseval equates the L2 norm on the circle to the sum of squared coefficients, and Jensen is applied to R itself. For arbitrary phases the partial sums need not converge in L2 or even pointwise, so R is not automatically an L2 function. The abstract gives no intermediate limiting step or Abel summation. If the full text supplies a rigorous passage from the formal series to the integral identities, the proof is fine; if it applies the theorems directly to the formal object, that step is missing. That is the only load-bearing point.

The work is narrow but honest. It targets readers already following unimodular series and the cited conjecture. The citation pattern is appropriate and the result is new relative to the referenced literature.

I would send it to a serious referee with the specific request to check the passage from formal series to the circle integrals. The conjecture is settled if that check passes.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that if R(z) = ∑ r_n z^n is a formal power series with |r_n| = 1 for every n ≥ 0, then for each integer m ≥ 2 the coefficient sequence of R(z)^m is unbounded. The argument combines Parseval's identity with Jensen's inequality and thereby confirms Conjecture 3.9 of Gawron, Miska, and Ulas.

Significance. If the central claim holds, the result settles an open conjecture on the growth of coefficients of powers of unimodular series and supplies a uniform obstruction to boundedness that applies to every choice of unimodular coefficients. The combination of two classical analytic tools to obtain a coefficient-wise conclusion is economical and potentially reusable.

major comments (2)

[Abstract and proof] Abstract (paragraph 1) and the proof section: Parseval's identity is invoked to equate the L² norm on the circle with the sum of squared coefficients of R(z)^m, yet the manuscript does not supply an intermediate limiting argument (Abel summation, Cesàro means, or weak-* convergence) that would justify the identity when the partial sums of R(e^{iθ}) fail to converge in L² for arbitrary phases. Without this step the application to a general formal series with |r_n|=1 remains formally unsupported.
[Proof] Proof (Jensen step): Jensen's inequality is applied directly to log |R(re^{iθ})| or an analogous quantity; the manuscript must clarify whether this is performed on the formal product before taking r→1 or on a truncated polynomial, and whether the resulting lower bound on the constant term of R^m survives the limit.

minor comments (2)

[Theorem statement] The statement of the main theorem should explicitly quantify over all sequences with |r_n|=1 rather than leaving the quantifier implicit.
[Introduction] A short remark on the relation between the formal power-series product and the analytic function on |z|<1 would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where additional justification is needed to make the analytic arguments fully rigorous for formal power series. We respond to each major comment below.

read point-by-point responses

Referee: [Abstract and proof] Abstract (paragraph 1) and the proof section: Parseval's identity is invoked to equate the L² norm on the circle with the sum of squared coefficients of R(z)^m, yet the manuscript does not supply an intermediate limiting argument (Abel summation, Cesàro means, or weak-* convergence) that would justify the identity when the partial sums of R(e^{iθ}) fail to converge in L² for arbitrary phases. Without this step the application to a general formal series with |r_n|=1 remains formally unsupported.

Authors: We agree that an explicit limiting argument is required, as the series with |r_n|=1 does not define an L² function on the circle. In the revision we will insert a preliminary step that applies Parseval to the Abel means R(r e^{iθ}) for r<1 (which are holomorphic and hence in L²), obtains the identity for the squared coefficients of the m-th power, and passes to the limit r→1^-. Because the coefficients themselves are independent of r, the resulting lower bound on the ℓ² norm of the coefficient sequence of R^m carries over directly to the formal series. revision: yes
Referee: [Proof] Proof (Jensen step): Jensen's inequality is applied directly to log |R(re^{iθ})| or an analogous quantity; the manuscript must clarify whether this is performed on the formal product before taking r→1 or on a truncated polynomial, and whether the resulting lower bound on the constant term of R^m survives the limit.

Authors: We will clarify that Jensen's inequality is applied to the Abel means (or to the polynomials obtained by truncating the series) inside the disk |z|=r<1. The resulting lower bound on the integral of log |R(r e^{iθ})|^m is independent of r and therefore persists as r→1^-. We will add a short paragraph spelling out this order of operations and invoking the monotone convergence of the means to justify that the lower bound on the constant term of R^m survives the limit. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on external standard theorems

full rationale

The paper's central claim is established by combining Parseval's identity and Jensen's inequality, both external mathematical results independent of the present work. No self-citations are load-bearing, no parameters are fitted then renamed as predictions, and no ansatz or uniqueness result is imported from prior author work. The argument is self-contained against external benchmarks and does not reduce any step to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the applicability of two standard analytic theorems to unimodular series; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

axioms (2)

standard math Parseval's identity applies to the power series on the unit circle
Invoked in the proof per abstract.
standard math Jensen's inequality applies to the convex function arising from the coefficient extraction
Invoked in the proof per abstract.

pith-pipeline@v0.9.1-grok · 5600 in / 1299 out tokens · 59169 ms · 2026-06-30T01:33:42.197192+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

1 extracted references

[1]

Gawron, P

[1] M. Gawron, P. Miska, M. Ulas,Arithmetic properties of coefficients of power series expansion of ∏∞ n=0(1− x2n )t, with an appendix by A. Schinzel, Monatsh. Math.185(2018), 307–360. School of Mathematics and Statistics, Central South University, Changsha, Hunan, China Email address:sz1021@csu.edu.cn

2018

[1] [1]

Gawron, P

[1] M. Gawron, P. Miska, M. Ulas,Arithmetic properties of coefficients of power series expansion of ∏∞ n=0(1− x2n )t, with an appendix by A. Schinzel, Monatsh. Math.185(2018), 307–360. School of Mathematics and Statistics, Central South University, Changsha, Hunan, China Email address:sz1021@csu.edu.cn

2018