A class of Littlewood polynomials that are not $L^α$-flat

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proposition_3_5: El Abdalaoui's proposition that a sequence of L2-normalized plus-or-minus-one polynomials that is L^alpha-flat for some alpha strictly between 0 and 2 has limiting frequency of the coefficient -1 in the closed interval from 1/4 to 3/4.

theorem_2_1: El Abdalaoui's theorem that a sequence of L2-normalized plus-or-minus-one polynomials whose limiting frequency of the coefficient -1 is not in the open interval (1/4, 3/4) is not L^alpha-flat for any alpha at least 0.

theorem_2_2: El Abdalaoui's theorem that a sequence of L2-normalized plus-or-minus-one polynomials whose limiting frequency of the coefficient -1 is not 1/2 has L^alpha norms tending to infinity, and so is not L^alpha-flat, for every alpha above 2.

theorem_2_3: El Abdalaoui's theorem that a sequence of L2-normalized plus-or-minus-one polynomials, each palindromic of even degree, is not L^alpha-flat for any alpha at least 0.

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E. H. el Abdalaoui, M. G. Nadkarni, "A class of Littlewood polynomials that are not $L^α$-flat," arXiv:1606.05852 (2016; v3, 2017). The paper is by el Abdalaoui; its appendix (Section 5) is written jointly with Nadkarni.

The copy read for this card is the arXiv text of arXiv:1606.05852v3 (10 May 2017), read in full; pages below are that PDF's own. Read status: claims checked for the definitions (pp. 4-6) and the statements of Theorems 2.1-2.3 (p. 7), Propositions 3.3 (p. 8) and 3.5 (p. 10), Theorem 3.6 (p. 10) and Theorem 4.1 (p. 11); their proofs were read for orientation but were not verified here. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1606.05852), every other right reserved.

Result pages: Theorem 2.1 (p. 7), Theorem 2.2 (p. 7), Theorem 2.3 (p. 7), Proposition 3.5 (p. 10).

Normalization

The paper writes an $L^2$-normalized Littlewood polynomial as (2.1), p. 4,

$$P_q(z)=\frac1{\sqrt q}\sum_{j=0}^{q-1}\epsilon_jz^j, \qquad \epsilon_j\in\{-1,1\},$$

and assumes (p. 5) that the limiting negative-coefficient frequency exists:

$$\operatorname{fr}(-1) =\lim_{q\to\infty}\frac1q \#\{0\le j<q:\epsilon_j=-1\}.$$

Thus $q$ is the number of coefficients and the degree is $q-1$. For $\alpha>0$ or $\alpha=+\infty$, $L^\alpha$-flat means $\lvert P_q\rvert\to1$ in $L^\alpha(S^1)$; for $\alpha=0$ it means that the Mahler measures tend to $1$ (Section 2, "Flat polynomials", p. 6).

Coefficient-frequency obstructions

• Proposition 3.5 (p. 10). If $(P_q)$ is $L^\alpha$-flat for some $0<\alpha<2$, then $\tfrac14\le\operatorname{fr}(-1)\le\tfrac34$.
• Theorem 2.2 (p. 7). If $\operatorname{fr}(-1)\ne\tfrac12$, then $(P_q)$ is not $L^\alpha$-flat for any $\alpha>2$, and $\lVert P_q\rVert_\alpha\to+\infty$. Its case $\alpha=4$ is the Jensen--Jensen--Høholdt theorem, which the paper quotes as Theorem 3.6 (p. 10) and presents as following from Theorem 2.2. The paper adds (p. 11) that a sequence flat in Littlewood's sense has frequency of $-1$ equal to $\tfrac12$.
• Theorem 2.1 (statement p. 7; proof on pp. 7-10 of Section 3, pp. 7-11, and the appendix written jointly with Nadkarni, Section 5, pp. 12-14). If $\operatorname{fr}(-1)\notin\,]\tfrac14,\tfrac34[$, including either endpoint, then $(P_q)$ is not $L^\alpha$-flat for any $\alpha\ge0$. Proposition 3.3 (p. 8) supplies the closed-interval necessary condition for $L^1$-flatness; the appendix treats the endpoint frequencies $\tfrac14$ and $\tfrac34$.

Symmetry obstruction

Theorem 2.3 (statement p. 7; proof in Section 4, pp. 11-12). If every $P_q$ is palindromic and has even degree, then the sequence is not $L^\alpha$-flat for any $\alpha\ge0$. Here palindromic means, for a degree $n$ polynomial, $\epsilon_j=\epsilon_{n-j}$ for all $j$ (p. 4). The proof rewrites the polynomial as a cosine polynomial and invokes Littlewood's flatness criterion (Theorem 4.1, p. 11).

Scope for Problem 1150

These are restricted obstructions, not a solution of the unrestricted maximum-modulus problem. A sequence of $\pm1$ polynomials with maxima at most $(1+o(1))\sqrt n$ has bounded normalized $L^\alpha$ norms, so by Theorem 2.2 its frequency of $-1$ tends to $\tfrac12$; and, being $L^\alpha$-flat for finite $\alpha$, it has by Theorem 2.3 only finitely many even-degree palindromic members. The paper gives no universal constant $c$ for all Littlewood polynomials, no quantitative value of such a gap, and no obstruction for the remaining balanced, non-palindromic family (nor for odd-degree palindromic or other symmetry classes not named by Theorem 2.3).

In particular, these 2016 coefficient-frequency and symmetry theorems should not be conflated with el Abdalaoui's later [[polynomials/abdalaoui_2025_l_alpha_flatness_erdos_littlewood_s/_index|claimed unrestricted $L^\alpha$-flatness result]], whose claimed resolution its own card records as contradicted. That later claim does not enlarge the scope of the results extracted here.

Bears on.

• #1150: a restriction only. Theorems 2.2 and 2.3 confine any sequence with maxima at most $(1+o(1))\sqrt n$ to the balanced cases outside the even-degree palindromic class, as just described; the paper does not settle the problem.
• #228: a necessary condition only. By the remark after Theorem 2.2 (p. 11), a sequence flat in Littlewood's sense has frequency of $-1$ equal to $\tfrac12$; the paper does not address whether such sequences exist.

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