The sequence of partial sums of a unimodular power series is not ultraflat

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Tamás Erdélyi, The sequence of partial sums of a unimodular power series is not ultraflat, arXiv:2504.19336 (2025), manuscript dated 23 April 2025.

The retained complete Markdown reading copy is a transcription of the five-page manuscript; it carries no page markers. The PDF is held locally. The source is available at https://arxiv.org/abs/2504.19336. The arXiv record (https://arxiv.org/abs/2504.19336, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Reading depth is claims checked for Theorem 2.1: its statement and proof on pp. 3--4 were read clause by clause to record the mechanism, including Lemma 2.2 as quoted and equations (2.1)--(2.3), but the proof was not independently verified. The proof of the imported Bernstein-factor lemma in Erdélyi's earlier paper [Er1] was not checked.

Theorem 2.1

Statement (Theorem 2.1, p. 3). If $(a_j)_{j=0}^{\infty}$ is a sequence of complex numbers with $|a_j|=1$ and

$$P_n(z)=\sum_{j=0}^n a_jz^j, \qquad n=0,1,2,\ldots,$$

then the partial sums $(P_n)$ do not form an ultraflat sequence in the sense of Definition 1.3.

Proof mechanism (pp. 3--4). Assuming the partial sums are ultraflat, the proof forms the reversed polynomials

$$Q_{n+1}(z)=1+z^{n+1}P_n(1/z).$$

They are again ultraflat, and direct coefficient summation gives the identity

$$\sum_{k=0}^n P_k(z)=z^nQ'_{n+1}(1/z). \tag{2.1}$$

The ultraflat upper bound and the triangle inequality make the left-hand side at most $\frac23(1+o(1))(n+2)^{3/2}$ uniformly on the unit circle; this is (2.2). The second equality of the Bernstein-factor Lemma 2.2 says that an ultraflat degree-$m$ polynomial has $\max|Q'_m|/\max|Q_m|=m+o(m)$. Applied to $Q_{n+1}$ and combined with (2.1), it supplies points where the same sum is at least $\frac34(n+2)^{3/2}$ for all sufficiently large $n$, as in (2.3). The two bounds contradict each other.

Scope for Problem 1150

The theorem excludes a nested construction: one must choose a single infinite unimodular coefficient sequence and use its initial segment at every degree. Its hypothesis already permits arbitrary complex phases, so it applies in particular to a single infinite plus-or-minus-one sequence.

Problem 1150, however, asks for one fixed $c>0$ giving a maximum-modulus gap for every plus-or-minus-one polynomial of every sufficiently large degree. It allows the coefficients to be redesigned at each degree, and it asks only about the maximum rather than two-sided uniform flatness. Thus Theorem 2.1 rules out the nested infinite-sequence route to ultraflatness but does not prove the universal gap required by E1150.

Bears on. #1150