Sharp Strichartz estimate for the 1D periodic Schrödinger equation

We prove the following estimate $$\mathrm{\parallel }{e}^{it{\mathrm{\partial }}_x^2}f{\mathrm{\parallel }}_{L_{(t,x)\in {\mathbb{T}}^2}^6}\le C(\log N{)}^{1\mathrm{/}6}\mathrm{\parallel }f{\mathrm{\parallel }}_{L_x^2(\mathbb{T})},$$ assuming $supp(\widehat{f})\subset [-N,N]$ for $N>1$. The bound $(\log N{)}^{1\mathrm{/}6}$ is sharp in view of the lower bound by Bourgain \cite{Bourgain}.