The L_4 norm of Littlewood polynomials derived from the Jacobi symbol

TL;DR

This paper investigates the growth of the L_4 norm of Littlewood polynomials derived from the Jacobi symbol, revealing that for prime degrees they have minimal growth, while for composite degrees the norm can vary widely.

Contribution

It introduces a new class of Littlewood polynomials based on the Jacobi symbol and analyzes their L_4 norm behavior, especially distinguishing prime and composite degrees.

Findings

For prime n, the normalized L_4 norm is asymptotically (7/6)^{1/4}.

For composite n, the normalized L_4 norm can grow arbitrarily large.

Proper coefficient choices can minimize the normalized L_4 norm to the known optimal value.

Abstract

Littlewood raised the question of how slowly the L_4 norm ||f||_4 of a Littlewood polynomial f (having all coefficients in {-1,+1}) of degree n-1 can grow with n. We consider such polynomials for odd square-free n, where \phi(n) coefficients are determined by the Jacobi symbol, but the remaining coefficients can be freely chosen. When n is prime, these polynomials have the smallest known asymptotic value of the normalised L_4 norm ||f||_4/||f||_2 among all Littlewood polynomials, namely (7/6)^{1/4}. When n is not prime, our results show that the normalised L_4 norm varies considerably according to the free choices of the coefficients and can even grow without bound. However, by suitably choosing these coefficients, the limit of the normalised L_4 norm can be made as small as the best known value (7/6)^{1/4}.