On a problem due to Littlewood concerning polynomials with unimodular coefficients

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.