# On $L^\alpha$-flatness of Erdös-Littlewood's polynomials.

El Abdalaoui El Houcein

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El Abdalaoui El Houcein. On $L^\alpha$-flatness of Erdös-Littlewood's polynomials.. 2025. hal-05052198

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Distributed under a Creative Commons Attribution 4.0 International License
ON $L^\alpha$-FLATNESS OF ERDÖS-LITTLEWOOD'S POLYNOMIALS.

el Houcein el Abdalaoui*

Begin at the beginning, the King
said gravely, “and go on till you
come to the end: then stop.”

Lewis Carroll, *Alice in Wonderland*

Those who know do not speak;
those who speak do not know.

Laozi (Lao Tzu)¹ª

ABSTRACT. It is shown that Erdős-Littlewood's polynomials are not $L^\alpha$-flat when $\alpha > 2$ is an even integer (and hence for any $\alpha \ge 4$). This provides a partial solution to an old problem posed by Littlewood. Consequently, we obtain a positive answer to the analogous Erdös-Newman conjecture for polynomials with coefficients $\pm 1$; that is, there is no ultraflat sequence of polynomials from the class of Erdős-Littlewood polynomials.

Our proof is short and simple. It relies on the classical lemma for $L^p$ norms of the Dirichlet kernel, the Marcinkiewicz-Zygmund interpolation inequalities, and the $p$-concentration theorem due to A. Bonami and S. Révész.

# 1. INTRODUCTION

Let $\mathcal{L}$ be the class of analytic trigonometric polynomials of the form

$$P_q(\theta) = \frac{1}{\sqrt{q}} \sum_{j=0}^{q-1} \epsilon_k e^{ik\theta},$$

where $\epsilon_k = \pm 1$. This class is said to be the class of Littlewood. It may be said also the class of Erdös-Littlewood (see [18, 19, 16]).

Here, we are interested on the behavior of those polynomials. Precisely on the $L^\alpha$-flatness, $\alpha \ge 0$ of a sequence $(P_n)$ from $\mathcal{L}$. As mentioned above, this problem arises in pure mathematics [18, 19, 16], and it turns out that it also arises in several

<footnote>ªThis Taoist idea can be rephrased à la Erdös's as : "Everyone writes, nobody reads."</footnote>

<footnote>Date: April 30, 2025.</footnote>

<footnote n="2020">2020 Mathematics Subject Classification. Primary 42A05, 42A55, Secondary 37A05, 37A30.</footnote>

<footnote>Key words and phrases. flat polynomials, ultraflat polynomials, Erdö-Littlewood's problem, Dirichlet kernel, Marcinkiewicz-Zygmund's interpolation inequalities, Simple Lebesgue component spectrum, Banach's problem, Banach-Rokhlin's problem, weak Rokhlin's problem.</footnote>
engineering problems [25, 26, 22].

A sequence $(P_q)$ of analytic trigonometric polynomials of class $\mathcal{L}$ is said to be $L^\alpha$-c-flat if $|P_q|$ converge to a constant $c$ with respect to $L^\alpha$-norm. Formally,

$$ (1) \qquad \left\| |P_q(z)| - c \right\|_{\alpha} \xrightarrow{q \to +\infty} 0. $$

If $c = 1$ the sequence $(P_q)$ is said to be $L^\alpha$-flat.

As it is customary, let us denote the torus by $\mathbb{T} = \{z \in \mathbb{C} : |z| = 1\}$, which can be identified with $[0, 1)$, $[-\frac{1}{2}, \frac{1}{2})$, $[0, 2\pi)$, or $[-\pi, \pi)$. Obviously, $L^\alpha$-flatness implies $L^\beta$-flatness, for any $\beta \le \alpha$, since for any polynomials $P$, $\|P\|_\beta \le \|P\|_\alpha$.

In this note, we present a straightforward and short proof showing that Erdös-Littlewood polynomials are not $L^\alpha$-flat when $\alpha$ is an even integer strictly greater than 2. This provides a positive answer to the Erdős conjecture (Problem 22 in [16]) for the class $\mathcal{L}$, extending the result in [3] and also supporting the numerical computations by A. Odlyzko [22].

It turns out that J. Bourgain and M. Guenais established a connection between the $L^1$-flatness of Erdős-Littlewood polynomials and a long-standing problem in the spectral theory of dynamical systems attributed to Banach and Rokhlin [13, 14]. This problem concerns the existence of a measure-preserving ergodic transformation on a probability measure space with a simple Lebesgue spectrum. Let us recall that the original Banach problem asked whether there exists a map acting on $\mathbb{R}$ with a simple Lebesgue spectrum. J. Bourgain linked this problem to the problem of $L^1$-flatness in the class of polynomials nodays known as Bourgain-Newman polynomials [5, 7, 13].

In [2], the author provided a positive answer to the original Banach problem by constructing an ergodic transformation acting on an infinite measure space with a simple Lebesgue spectrum (see also [4] for a simpler proof). However, the author emphasized therein that the $L^1$-flatness of Erdős-Littlewood polynomials remains an open question.

It should be noted that our work does not provide any insight into the $L^\alpha$-flatness problem for Erdős-Littlewood polynomials when $\alpha < 2$. Furthermore, we do not believe that our method is applicable in this case.

## 2. MAIN RESULT AND ITS PROOF.

In this section, we will state and prove our main result.

**Theorem 1.** [Theorem of El Roc de Sant Gaietà<fnref n="1" />] There is no sequence from the class $\mathcal{L}$ which is $L^{2p}$-flat, for any positive integer $p > 1$.

Consequently, we have,

**Corollary 1.** There is no sequence from the class $\mathcal{L}$ which is $L^\alpha$-flat, for any $\alpha \ge 4$.

*Proof.* Assume that there is a sequence from the class $\mathcal{L}$ which is $L^\alpha$-flat, for $\alpha \ge 4$. There this sequence is $L^{2p}$-flat, where $p = \lfloor \frac{\alpha}{2} \rfloor$, which contradicts our main result. □

<footnote n="1">¹Bachelard points out that places have a profound effect on our imagination and can inspire ideas and works [13]. So, it can be suggested to name theorems after specific places.!</footnote>
We further have

**Corollary 2.** *The Erdös conjecture on the existence of ultraflat from the class L is true, that is, there is no ultraflat sequence of polynomials (P<sub>q</sub>) ⊂ L.*

*Proof.* Assume by contradiction that such a sequence exists. Then this sequence is L<sup>α</sup>-flat for any α > 0. This contradicts our main result and the proof is complete.

<figure><img src="image_2.png" /></figure>

For the proof of Theorem 1, we need the following classical lemma on the computation of L<sup>p</sup> norms of Dirichlet Kernel. Its proof can be founded here [1].

**Lemma 1** (*L*<sup>*p*</sup>-norm of Dirichlet Kernel). Let *N* be a positive integer and

$$
D_N(z) = \sum_{j=0}^{N-1} z^j.
$$

Then, for any $p > 1$,

$$
\|D_N\|_p^p = \delta_p N^{p-1} + R_p(N^{p-1}), \text{ as } N \to +\infty,
$$

where

$$
\delta_p = \frac{2}{\pi} \int_0^\infty \left| \frac{\sin(x)}{x} \right|^p dx,
$$

and

$$
R_p(N) = \begin{cases} O_p(N^{p-3}) & \text{if } p > 3, \\ O_p(\ln(N)) & \text{if } p = 3, \\ O_p(1) & \text{if } 1 \le p < 3, \end{cases}
$$

We need also the following observation form [8] (see eq. (2.2) and (2.3))

$$
(2) \quad P_q(z) = \frac{1}{\sqrt{q}} D_q(z) - \frac{2}{\sqrt{q}} \sum_{j=0}^{N-1} \eta_j' z^j
$$

$$
(3) \quad = \frac{2}{\sqrt{q}} \sum_{j=0}^{N-1} \eta_j z^j - \frac{1}{\sqrt{q}} D_q(z),
$$

$$
(4) \quad \text{where } e^{i\theta} = z, \theta \in \mathbb{R}, \text{ and } \eta_j, \eta'_j \in \{0,1\}, j = 0, \dots, N-1.
$$

We recall also the following lemma from [2] which is related to the so-called L⁴-
strategy due D.J. Newman & Byrnes [21].

**Lemma 2** (*L*<sup>4</sup>-norm of Bourgain-Newmann polynomials). Let *q* be a positive integer and

$$
Q_q(z) = \sum_{j=0}^{q-1} \eta_j e^{2\pi j x},
$$

where $x \in [0,1)$, $\eta_j \in \{0,1\}$, $j = 0, \dots, q-1$. Then,

$$
\liminf_{q \to +\infty} \frac{\|Q_q\|_4}{\|Q_q\|_2} \ge 2.
$$
From Lemma 2, it can be shown that the Newmann-Bourgain polynomials are not $L^\alpha$-flat for $\alpha \ge 4$ (see [6]). In a forthcoming paper, we will use it again to prove that Ben Green's polynomials are not $L^\alpha$-flat for $\alpha > 0$.

**Proposition 1.** Let $(Q_q)$ be a sequence of Bourgain-Newmann polynomials and assume that the density of 1 is positive. Then, for any $\alpha > 2$,

$$ \frac{\|Q_q\|_{\alpha}}{\|Q_q\|_2} \xrightarrow{q \to +\infty} +\infty. $$

In addition, we require the following lemma, which corresponds to the Marcinkiewicz-Zygmund interpolation inequalities [27, Theorem 2.7, Chap. X, Vol. II, p. 30].

**Lemma 3.** [Marcinkiewicz-Zygmund interpolation inequalities] Let $\alpha \in [1, +\infty]$. Then, for any analytic polynomial $P$ of degree at most $n-1$, there are two positive constants $A'$, $A'_\alpha$ such that

$$ (5) \qquad \left( \frac{1}{n} \sum_{j=0}^{n-1} |P(\xi_{j,n})|^\alpha \right)^{\frac{1}{\alpha}} \le A' \cdot \|P(z)\|_{\alpha} \quad \text{if } \alpha \in [1, +\infty], $$

$$ (6) \qquad \|P(z)\|_{\alpha} \le \left( \frac{A'_{\alpha}}{n} \sum_{j=0}^{n-1} |P(\xi_{n,j})|^{\alpha} \right)^{\frac{1}{\alpha}}, \quad \text{if } \alpha \in [1, +\infty[ $$

where $(\xi_{n,j})$ are the $n$-th root of unity given by

$$ \xi_{n,j} = e^{2\pi i \frac{j}{n}}, \quad j = 0, \dots, n-1. $$

According to Zygmund [27, vol II, Chap X, p. 31], the constants are defined as $A' = 2A + 1$ and $A'_\alpha = 2A_\alpha + 1$, where $A = \sup_{\alpha>1}(\pi\alpha + 1)^{\frac{1}{\alpha}}$ and $A_\alpha$ is a similar constant obtained for the Marcinkiewicz-Zygmund inequalities for trigonometric polynomials of degree at most $n$ (see also [20]). Moreover, the constant $A_\alpha$ does not exceed a fixed multiple, independent of $\alpha$, of the constant in the Riesz Theorem for the conjugate function [27, Vol I, Chap VII, p. 253]. The sharp constant in that theorem are

$$ A_{\alpha} = \begin{cases} \tan\left(\frac{\pi}{2\alpha}\right) & \text{if } 1 < \alpha \le 2, \\ \cot\left(\frac{\pi}{2\alpha}\right) & \text{if } 2 \le \alpha < +\infty \end{cases} $$

This later result is due to Pichorides. We refer to [17], for a survey on the subject.

Let us also recall the following lemma from [2]. For the sake of completeness, we will provide its proof. We believe that this lemma will find application in further research.

**Lemma 4** (Flatness implies zero density). Let $q$ be a positive integer, $c$ a positive number, $\alpha > 2$ and

$$ Q_q(x) = \sum_{j=0}^{q-1} \eta_j e^{2\pi j x}, $$

where $x \in [0, 1)$, $\eta_j \in \{0, 1\}$, $j = 0, \dots, q-1$. Assume that the sequence $(\frac{1}{\sqrt{q}} \cdot Q_q(x))$ is $L^\alpha$-c-flat. Then, the density of the set $S_q = \{0 \le j \le q-1 : \eta_j = 1\}$ vanish as $q \to +\infty$, that is,
$$ \frac{|S_q|}{q} \xrightarrow{q \to +\infty} 0. $$

*Proof.* For $\alpha > 1$, by Lemma 3, we have

$$ (7) \qquad \left\| \frac{1}{\sqrt{q}} \cdot Q_q(x) \right\|_{\alpha}^{\alpha} \ge A'_{\alpha} \cdot \frac{1}{q} \cdot \left| \frac{Q_q(1)}{\sqrt{q}} \right|^{\alpha} = A'_{\alpha} \frac{2^{\alpha}}{q^{1+\frac{\alpha}{2}}} \cdot q^{\alpha} \cdot \left( \frac{|S_q|}{q} \right)^{\alpha} $$

$$ (8) \qquad \ge A'_{\alpha} \frac{2^{\alpha}}{q^{1-\frac{\alpha}{2}}} \cdot \left( \frac{|S_q|}{q} \right)^{\alpha} $$

But, by our assumption $\alpha > 2$ and the sequence $(\frac{1}{\sqrt{q}} \cdot Q_q(x))$ is $L^\alpha$-c-flat. Therefore

$$ \left\| \frac{1}{\sqrt{q}} \cdot Q_q(x) \right\|_{\alpha}^{\alpha} \xrightarrow{q \to +\infty} c^{\alpha}, $$

and

$$ q^{1-\frac{\alpha}{2}} \xrightarrow{q \to +\infty} 0. $$

We thus conclude

$$ \frac{|S_q|}{q} \xrightarrow{q \to +\infty} 0, $$

and the lemma has been proven.

□

Under certain restricted conditions, the preceding proof enables us to improve the result in [6] as follows.

**Proposition 2.** Let $(Q_q)$ be a sequence of Bourgain-Newmann polynomials and assume that the density of 1 is positive. Then, for any $\alpha > 2$,

$$ \frac{\|Q_q\|_{\alpha}}{\|Q_q\|_2} \xrightarrow{q \to +\infty} +\infty. $$

*Proof.* Let $Q_q(z) = \sum_{j=0}^{q-1} \eta_j z^j$, where $z \in \mathbb{T}$, $\eta_j \in \{0, 1\}$, $j = 0, \dots, q-1$, and put

$$ S_q = \{0 \le j \le q-1 : \eta_j = 1\}. $$

Then, by our assumption

$$ \frac{|S_q|}{q} \xrightarrow{q \to +\infty} d > 0. $$

Applying Lemma 3, it follows

$$ (9) \qquad \left\| \frac{1}{\sqrt{|S_q|}} \cdot Q_q(z) \right\|_{\alpha}^{\alpha} \ge A'_{\alpha} \cdot \frac{1}{q} \cdot \left| \frac{Q_q(1)}{\sqrt{|S_q|}} \right|^{\alpha} $$

But $|S_q| \sim d.q.$ Therefore,

$$ (10) \qquad \left\| \frac{1}{\sqrt{|S_q|}} \cdot Q_q(z) \right\|_{\alpha}^{\alpha} \ge A'_{\alpha} \cdot \frac{1}{q} \cdot \left( \sqrt{|S_q|} \right)^{\alpha} $$

$$ (11) \qquad \ge c_{\alpha} q^{\frac{\alpha}{2}-1} \xrightarrow{q \to +\infty} +\infty, $$

since $\alpha > 2$. The proof of the proposition is complete.

□
We further need a crucial lemma from [10] which corresponds to one of the main results therein. Before stating the lemma, we introduce the following definition

Let $\mathcal{P}$ be the set of polynomials with coefficients 0 or 1. ( called also idempotents). For each $p > 0$, define $C_p$ as the largest number such that for every set $E$, $E \subset \mathbb{T}$ with $|E| > 0$, the inequality

$$ \sup_{P \in \mathcal{P}} \frac{\int_E |P(z)|^p dz}{\int |P(z)|^p dz} \geq C_p. $$

holds. $|E|$ denotes the Lebesgue measure of $E$ and the definition of $C_p$ is extended to the limit case $p = \infty$ in the usual way.

Following [11], For $p > 0$, we say that there is $p$-concentration if there exists a constant $c > 0$ so that for any symmetric non empty open set $E$ one can find an idempotent $P \in \mathcal{P}$ with

$$ (12) \qquad \int_E |P(z)|^p dz \geq c \int |P(z)|^p dz. $$

Moreover, $c_p$ will denote the supremum of all such constants $c$. Correspondingly, $c_p$ is called the level of $p$-concentration. If $c_p = 1$, we say that there is full $p$-concentration. In [11], the authors proved the following.

**Lemma 5.** For all $0 < p < \infty$ we have $p$-concentration. Moreover, if $p$ is not an even integer, then we have full concentration, i.e. $c_p = 1$. When considering even integers, we have $c_2 = \sup_{x \ge 0} \frac{\sin(x)^2}{\pi x}$, then $0.495 < c_4 \le 1/2$, then for all other even integers $0.483 < c_{2k} \le 1/2$.

The previous lemma addressed the $p$-concentration inequalities (see [10, 11] for details). The constant $c_2$ is due to Déchamps-Gondim, Lust-Piquard and Queffélec [15]. It is also related to the classical Weiner-Shapiro inequality [24] (see also [12]).

We now proceed with the proof of our main theorem (Theorem 1).

*Proof.* By (2), we have

$$ (13) \qquad P_q(z) = \frac{1}{\sqrt{q}} (2Q_q(z) - D_q(z)), $$

where $Q_q(z) = \sum_{j=0}^{q-1} \eta_j z^j$ and $\eta_j \in \{0, 1\}, j = 0, \dots, q-1$. It follows that for any $q$-th root of unity $\xi_{j,q}, j = 1, \dots, q-1$, we can write

$$ (14) \qquad P_q(\xi_{j,q}) = \frac{-2}{\sqrt{q}} Q_q(\xi_{j,q}) $$

Let us assume that $(P_q)$ is $L^\alpha$-flat for $\alpha > 2$. Then, we have

$$ \int ||P_q(z)| - 1||^\alpha dz \xrightarrow{q \to +\infty} 0. $$

We can thus extract a subsequence which we still denote by $(P_q)$ such that for almost all $z \in \mathbb{T}$, we have

$$ |P_q(z)| \xrightarrow{q \to +\infty} 1. $$
Hence, for almost all $z \in \mathbb{T}$,

$$ \left| \frac{2}{\sqrt{q}} Q_q(z) \right| \xrightarrow{q \to +\infty} 1. $$

Since, for any $z \neq 1$,

$$ \frac{|D_q(z)|}{\sqrt{q}} \xrightarrow{q \to +\infty} 0. $$

Therefore, by applying Vitali's theorem [8], for any $\beta < 2$,

$$ \left\| \frac{2}{\sqrt{q}} Q_q(z) - 1 \right\|_{\beta} \xrightarrow{q \to +\infty} 0. $$

Morover, by the triangle inequalities combined with Lemma 1, we have

$$ (15) \qquad \left| \frac{\left\| \frac{2}{\sqrt{q}} Q_q(z) \right\|_{\alpha}}{\left\| \frac{D_q(z)}{\sqrt{q}} \right\|_{\alpha}} - 1 \right| \le \frac{\left\| P_q \right\|_{\alpha}}{\left\| \frac{D_q(z)}{\sqrt{q}} \right\|_{\alpha}} \xrightarrow{q \to +\infty} 0. $$

Whence

$$ (16) \qquad \frac{\left\| 2 \cdot Q_q(z) \right\|_{\alpha}}{\left\| D_q(z) \right\|_{\alpha}} \xrightarrow{q \to +\infty} 1. $$

Now, apply Lemma 3 to the analytic polynomial $\frac{2}{\sqrt{q}} \cdot Q_q$ to get

$$ (17) \qquad \left\| \frac{2}{\sqrt{q}} Q_q(z) \right\|_{\alpha}^{\alpha} \le A'_{\alpha} \left( \frac{1}{q} \cdot \left| \frac{2}{\sqrt{q}} Q_q(1) \right|^{\alpha} + \frac{1}{q} \sum_{j=1}^{q-1} \left| \frac{1}{\sqrt{q}} \cdot Q(\xi_{j,q}) \right|^{\alpha} \right) $$

We claim that we can assume without loss of generality that the sequence

$$ \left( \frac{1}{q} \sum_{j=1}^{q-1} \left| \frac{1}{\sqrt{q}} \cdot Q(\xi_{j,q}) \right|^{\alpha} \right)_{q \ge 1} $$

converge to some constante $C_{\alpha}$. Indeed, by applying once again Lemma 3 to the analytic polynomial $P_q$, we can write

$$ (18) \qquad A'_{\alpha} \cdot \frac{1}{q} \sum_{j=0}^{q-1} \left| P_q(\xi_{j,q}) \right|^{\alpha} \le \left\| P_q \right\|_{\alpha}. $$

But under our assumption, for $\alpha > 2$, we have

$$ \int \left| P_q(z) - 1 \right|^{\alpha} dz \xrightarrow{q \to +\infty} 0. $$

Therefore,

$$ \left\| P_q \right\|_{\alpha} \xrightarrow{q \to +\infty} 1. $$

We thus deduce that the sequence

$$ \left( \frac{1}{q} \sum_{j=0}^{q-1} \left| P(\xi_{j,q}) \right|^{\alpha} \right)_{q \ge 1} $$
is a bounded sequence. So that we can extract a convergent subsequence from it.
But, from (14), we have

$$ \frac{1}{q} \sum_{j=0}^{q-1} |P(\xi_{j,q})|^\alpha = \frac{1}{q} \sum_{j=1}^{q-1} \left| \frac{2}{\sqrt{q}} \cdot Q(\xi_{j,q}) \right|^\alpha. $$

We can thus assume without loss of generality that it converges to some positive constant $C_\alpha$. This proved the claim. We still need to estimate

$$ \frac{1}{q} \cdot \left| \frac{2}{\sqrt{q}} Q_q(1) \right| $$

For that notice that we have

$$ Q_q(1) = \sum_{j=0}^{q-1} \eta_j = |S_q|, $$

and by appealing once again to Lemma 3, it can shown that the density of $S_q$ converge to $\frac{1}{2}$, that is,

$$ (19) \qquad \frac{|S_q|}{q} \xrightarrow{q \to +\infty} \frac{1}{2}. $$

Furthermore, for any $\delta > 0$, for any mesurable set $E_\delta \subset (-\delta, +\delta)^c$, it easy to see that

$$ (20) \qquad \int_{E_\delta} |P_q(\theta)|^\alpha d\theta \xrightarrow{q \to +\infty} |E_\delta|, $$

where $|E_\delta|$ is the Lebesgue measure of $E_\delta$. But, it is well known that

$$ (21) \qquad \int_{(-\delta, +\delta)^c} \frac{1}{q^{\frac{1}{\alpha}}} |D_q(e^{ix})|^\alpha dx \le \frac{2^\alpha}{\sqrt{q}} \frac{\sin(\delta/2)}{(\sin(\delta/2))^\alpha} dx \xrightarrow{q \to +\infty} 0. $$

Now, applying the triangle inequalities, we obtain

$$ (22) \qquad \left| \left( \int_{E_\delta} \left| \frac{2}{\sqrt{q}} \cdot Q_q(x) \right|^\alpha dx \right)^{\frac{1}{\alpha}} - \left( \int_{E_\delta} \left| \frac{1}{\sqrt{q}} \cdot D_q(e^{ix}) \right|^\alpha dx \right)^{\frac{1}{\alpha}} \right| $$

$$ (23) \qquad \le \left\| \mathbb{I}_{E_\delta} \cdot P_q(x) \right\|_{\alpha} \le \left( \int_{E_\delta} \left| \frac{2}{\sqrt{q}} \cdot Q_q(x) \right|^\alpha dx \right)^{\frac{1}{\alpha}} + \left( \int_{E_\delta} \left| \frac{1}{\sqrt{q}} \cdot D_q(e^{ix}) \right|^\alpha dx \right)^{\frac{1}{\alpha}} $$

Letting $q$ goes to infinity, we see

$$ (24) \qquad \int_{E_\delta} \left| \frac{2}{\sqrt{q}} \cdot Q_q(x) \right|^\alpha dx \xrightarrow{q \to +\infty} |E_\delta|. $$

The same reasoning yields

$$ \frac{\left\| \frac{2}{\sqrt{q}} \cdot Q_q \right\|_{\alpha}}{\left\| \frac{1}{\sqrt{q}} \cdot D_q \right\|_{\alpha}} = \frac{\left\| 2 \cdot Q_q \right\|_{\alpha}}{\left\| D_q \right\|_{\alpha}} \xrightarrow{q \to +\infty} 1. $$

From this combined with (24) and Lemma 1, we obtain

$$ (25) \qquad \frac{\int_{E_\delta} \left| \frac{2}{\sqrt{q}} \cdot Q_q(x) \right|^\alpha dx}{\left\| \frac{2}{\sqrt{q}} \cdot Q_q \right\|_{\alpha}} \xrightarrow{q \to +\infty} 0. $$
Whence

$$
(26) \quad \frac{\int \left| \frac{2}{\sqrt{q}} \cdot Q_q(x) \right|^\alpha \left| - \int_{(-\delta, \delta)} \left| \frac{2}{\sqrt{q}} \cdot Q_q(x) \right|^\alpha dx \right| \frac{2}{\sqrt{q}} \cdot Q_q \|_\alpha^\alpha}{\| \frac{2}{\sqrt{q}} \cdot Q_q \|_\alpha^\alpha} \xrightarrow{q \to +\infty} 0.
$$

In other words, we can write

$$
(27) \quad \frac{\int_{(-\delta, \delta)} \left| \frac{2}{\sqrt{q}} \cdot Q_q(x) \right|^\alpha dx}{\| \frac{2}{\sqrt{q}} \cdot Q_q \|_\alpha^\alpha} \xrightarrow{q \to +\infty} 1.
$$

Now, by taking $\alpha = 2p$, $p > 1$ a positive integer and applying Lemma 5, we get a contradiction. □

We conjecture the following

**Conjecture.** There is no $L^\alpha$-flat polynomials from the class of Erdős-Littlewood polynomials (class $\mathcal{L}$) for any $\alpha < 4$.

**Acknowledgment.** The author wishes to express his thanks to Jean-Marie Strelcyn for bring to his attention that the spectral Banach problem in ergodic theory can be found in Ulam’s book [23]. The author would like to express their heartfelt thanks to Michael Lin for the invitation to the University of Ben-Gurion, where this work was revisited.

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<footnote>* UNIVERSITY OF ROUEN NORMANDY, DEPARTMENT OF MATHEMATICS, LMRS UMR 60 85 CNRS, AVENUE DE L'UNIVERSITÉ, BP.12 76801 SAINT ETIENNE DU ROUVRAY - FRANCE.
Email address: elhoucein.elabdalaoui@univ-rouen.fr
URL: http://www.univ-rouen.fr/LMRS/Persopage/Elabdalaoui/</footnote>