On a problem due to Littlewood concerning polynomials with unimodular coefficients

Littlewood raised the question of how slowly $lVert f_{n}rVert_{4}^{4}-lVert f_{n}rVert_{2}^{4}$ (where $lVert.rVert _{r}$ 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 $(lVert g_{n}rVert_{4}^{4}-lVert g_{n}rVert_{2}^{4})/lVert g_{n}rVert_{2}^{3}$ is 2/π, which resolves a mystery due to Littlewood. This is however not the best answer to Littlewood’s question: for the polynomials $$h_n(z)=sum_{j=0}^{n-1}sum _{k=0}^{n-1} e^{2pi ijk/n} z^{nj+k} $$ the limit of $(lVert h_{n}rVert_{4}^{4}-lVert h_{n}rVert_{2}^{4})/lVert h_{n}rVert_{2}^{3}$ is shown to be 4/π 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.