Erdélyi: Ultraflat polynomials and conjugate reciprocals

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Tamás Erdélyi, "The asymptotic distance between an ultraflat unimodular polynomial and its conjugate reciprocal," arXiv:1810.04287 (2018).

The retained folder-name PDF is the arXiv version stamped "arXiv:1810.04287v2 [math.CA] 12 Feb 2019", the canonical version for this card; its conversion sits beside it. Provenance: downloaded from https://arxiv.org/pdf/1810.04287v2 on 2026-09-25; 191,332 bytes. The arXiv record (https://arxiv.org/abs/1810.04287, read 2026-10-02) names the CC0 1.0 Universal public domain dedication.

Bears on. E1150.

Overview

The paper studies the asymptotic separation of an ultraflat complex unimodular polynomial from its conjugate reciprocal. Here $K_n=\{\sum_{k=0}^n a_kz^k:|a_k|=1\}$, $P_n^*(z)=\sum_{k=0}^n\overline{a}_{n-k,n}z^k$, and ultraflatness means $|P_n(e^{it})|/\sqrt{n+1}\to1$ uniformly (Definitions 1.2–1.3). The introduction distinguishes this complex class from the Littlewood class $L_n$, recalls Kahane's cited construction of ultraflat sequences in $K_n$, and explicitly says that the analogous Erdős conjecture for $L_n$ remains unsettled (§1, especially (1.2)).

The principal result, Theorem 2.1, is the finite-moment asymptotic

$$\frac1{2\pi}\int_0^{2\pi}|(P_n-P_n^*)(e^{it})|^q\,dt \sim 2^qK(q)n^{q/2},\qquad K(q)=\frac{\Gamma((q+1)/2)}{\Gamma(q/2+1)\sqrt\pi},$$

for every fixed $q>0$. Its $q=2$ specialization is Theorem 2.2:

$$\sum_{k=0}^n|a_{k,n}-\overline a_{n-k,n}|^2 =\frac1{2\pi}\int_0^{2\pi}|P_n-P_n^*|^2\,dt\sim2n.$$

This sharpens the earlier lower bound recalled as Theorem 1.6. The derivative analogue, Theorem 2.3, gives

$$\sum_{k=0}^n k^2|a_{k,n}-\overline a_{n-k,n}|^2 =\frac1{2\pi}\int_0^{2\pi}|P_n'-P_n^{*\prime}|^2\,dt \sim\frac23n^3,$$

sharpening Theorem 1.5. Theorem 2.4 determines every fixed positive moment of the derivative of the modulus:

$$\frac1{2\pi}\int_0^{2\pi}\left|\frac d{dt}|(P_n-P_n^*)(e^{it})|\right|^qdt \sim\frac{K(q)}{q+1}n^{3q/2}.$$

Theorems 2.5 and 2.6 derive the near-orthogonality relations $\sum_{k=0}^n a_{k,n}a_{n-k,n}=o(n)$ and $\sum_{k=0}^n k^2a_{k,n}a_{n-k,n}=o(n^3)$; the first recovers Saffari's near-orthogonality conjecture, proved earlier in the paper's reference [Er4], and the second is new. Theorem 2.7 recovers previously known analogous moment formulas for $\operatorname{Re}P_n(e^{it})$ and its derivative. The remark following Theorem 2.7 states that all these assertions remain valid along arbitrary increasing subsequences of degrees.

The method is phase analysis. Writing $P_n(e^{it})=R_n(t)e^{i\alpha_n(t)}$ as in (1.3), the reciprocal phase satisfies $\alpha_n'+\alpha_n^{*\prime}=n$, equation (1.5). The central variable is

$$\beta_n(t)=\tfrac12(\alpha_n(t)-\alpha_n^*(t))=\alpha_n(t)-nt/2-t_0$$

((4.1)), so that $P_n-P_n^*$ is governed by $2R_n\sin\beta_n$ (Lemma 3.5). The proof imports the uniform distribution of $\alpha_n'/n$ from the earlier work cited in Lemma 3.1, uses the angular-speed bound (3.1) of Lemma 3.2, the higher-derivative estimate of Lemma 3.3, and the bound on the modulus derivative $R_n'$ in Lemma 3.4. Lemma 3.7 supplies the exact trigonometric mean $K(q)$. Lemmas 3.8 and 3.9 then establish oscillatory averaging by partitioning $[0,2\pi]$, locally linearizing $\beta_n$, and showing that the region of small phase speed is negligible; their hypotheses and conclusions are recorded in (3.2)–(3.9). These lemmas, combined respectively with Lemmas 3.5 and 3.6, prove Theorems 2.1 and 2.4 (§4).

For Theorem 2.3, uniform flatness gives (4.2); the cosine-law expansion (4.3), Parseval identity (4.4), integration-by-parts estimate (4.5), and vanishing average of $\cos(2\beta_n)$ in (4.6) yield the constant $2/3$. Theorems 2.5–2.6 follow from Theorems 2.2–2.3 by rotating $P_n(z)$ to $P_n(cz)$ and varying $|c|=1$. The results are asymptotic, restricted to finite $q$, and conditional on two-sided uniform ultraflatness; the paper neither constructs such sequences nor proves their impossibility in $L_n$. The locators above are the paper's theorem, lemma, equation, and section numbers.

Relation to E1150

For E1150 write

$$P_n(z)=\sum_{k=0}^n\varepsilon_{k,n}z^k, \qquad \varepsilon_{k,n}\in\{-1,1\}.$$

Then $P_n\in L_n\subset K_n$, and the paper's conjugate reciprocal becomes the ordinary reciprocal

$$P_n^*(z)=\sum_{k=0}^n\varepsilon_{n-k,n}z^k.$$

The distinction between $\sqrt n$ in E1150 and $\sqrt{n+1}$ in Definitions 1.2–1.3 is asymptotically negligible but should be retained in any exact formulation.

If a hypothetical Littlewood sequence were ultraflat in the paper's two-sided sense, Theorems 2.1–2.3 would impose strong necessary conditions. In particular, if $M_n=|\{k:0\le k\le n,\ \varepsilon_{k,n}\ne\varepsilon_{n-k,n}\}|$, then Theorem 2.2 gives

$$4M_n=\sum_{k=0}^n|\varepsilon_{k,n}-\varepsilon_{n-k,n}|^2\sim2n, \qquad M_n\sim n/2.$$

Thus approximately half of the ordered coefficient positions must disagree with their reversed partners. Theorem 2.3 further gives

$$4\sum_{\varepsilon_{k,n}\ne\varepsilon_{n-k,n}}k^2\sim\frac23n^3, \qquad \sum_{\varepsilon_{k,n}\ne\varepsilon_{n-k,n}}k^2\sim\frac16n^3.$$

Equivalently, Theorems 2.5–2.6 become

$$\sum_{k=0}^n\varepsilon_{k,n}\varepsilon_{n-k,n}=o(n), \qquad \sum_{k=0}^n k^2\varepsilon_{k,n}\varepsilon_{n-k,n}=o(n^3).$$

The full family in Theorem 2.1 additionally prescribes every fixed finite $L^q$-moment of the anti-reciprocal part $P_n-P_n^*$, while Theorem 2.4 prescribes moments of the derivative of its modulus. These facts could enter an E1150 contradiction argument after an independent step upgrades near-minimal maximum modulus to two-sided ultraflatness; they then provide phase-distribution and coefficient-reversal constraints against which the binary coefficients could be tested.

That upgrade is not supplied here. Negating E1150 yields, along a subsequence, Littlewood polynomials with $\max_{|z|=1}|P_n(z)|/\sqrt n\to1$, but this is only an upper-flatness assertion. Parseval fixes the mean square, yet does not by itself imply the uniform lower bound required by Definition 1.3. Moreover, the necessary reversal and correlation asymptotics above are not shown to be incompatible with $\{\pm1\}$-coefficients. The paper explicitly describes the $L_n$ analogue of (1.2) as unsettled (§1), and its cited discussion of Kahane concerns the larger complex class $K_n$. Consequently, the paper does not prove E1150, produce a Littlewood counterexample, or furnish an effective constant $c>0$; its relevance is as a structural description of what any genuinely ultraflat Littlewood subsequence would have to satisfy.