On a problem due to Littlewood concerning polynomials with unimodular coefficients

TL;DR

This paper investigates the growth of the difference between the fourth and second norms of polynomials with unimodular coefficients, providing explicit limits for specific sequences and highlighting open questions in the field.

Contribution

The paper computes explicit limits for the growth of ||f_n||_4^4 - ||f_n||_2^4 for particular polynomial sequences, resolving a Littlewood question and identifying open problems.

Findings

Limit for g_n(z) sequence is 2/π.

Limit for h_n(z) sequence is 4/π^2.

Open problem: existence of polynomials with larger growth.

Abstract

Littlewood raised the question of how slowly ||f_n||_4^4-||f_n||_2^4 (where ||.||_r denotes the L^r norm on the unit circle) can grow for a sequence of polynomials f_n with unimodular coefficients and increasing degree. The results of this paper are the following. For g_n(z)=\sum_{k=0}^{n-1}e^{\pi ik^2/n} z^k the limit of (||g_n||_4^4-||g_n||_2^4)/||g_n||_2^3 is 2/\pi, which resolves a mystery due to Littlewood. This is however not the best answer to Littlewood's question: for the polynomials h_n(z)=\sum_{j=0}^{n-1}\sum_{k=0}^{n-1} e^{2\pi ijk/n} z^{nj+k} the limit of (||h_n||_4^4-||h_n||_2^4)/||h_n||_2^3 is shown to be 4/\pi^2. No sequence of polynomials with unimodular coefficients is known that gives a better answer to Littlewood's question. It is an open question as to whether such a sequence of polynomials exists.