On $L^α$-flatness of Erdős-Littlewood's polynomials

TL;DR

This work addresses the problem of $L^\alpha$-flatness for Erdős-Littlewood polynomials, showing that no sequence in this class can be $L^{\alpha}$-flat when $\alpha$ is an even integer greater than $2$ (and thus for all $\alpha\ge4$). By establishing a short, self-contained argument that leverages $L^p$-norms of the Dirichlet kernel, Marcinkiewicz–Zygmund interpolation inequalities, and Bonami–Révész $p$-concentration, the authors deduce a strong non-flatness result and consequently a positive answer to the Erdős–Newman conjecture for these $\pm1$ coefficient polynomials: there is no ultraflat sequence in this class. The proof also employs a flatness-implies-zero-density lemma and connects to foundational questions in spectral theory via the Banach–Rokhlin problem. Overall, the paper provides a concise analytic route to a classical flatness problem with implications for ergodic theory and spectral questions.

Abstract

It is shown that Erdös--Littlewood's polynomials are not $L^α$-flat when $α> 2$ is an even integer (and hence for any $α\geq 4$). 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 Erdös--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.

Theorems & Definitions (16)

• Theorem 1
• Corollary 1
• proof
• Corollary 2
• proof
• Lemma 1: $L^p$-norm of Dirichlet Kernel
• Lemma 2: $L^4$-norm of Bourgain-Newmann polynomials
• Proposition 1
• Lemma 3
• Lemma 4: Flatness implies zero density