Finite Toeplitz matrices and sharp Littlewood conjectures

The sharp Littlewood conjecture states that for fixed $N\ge1$, if $D(z)=1+z+z^2+\dots+z^{N-1}$, then on the unit circle $|z|=1$, $\|D\|_1$ is the minimum of $\|f\|_1$ for $f$ of the form $f(z)=c_0+c_1z^{n_1}+\dots+c_{N-1}z^{n_{N-1}}$ with $|c_k|=1$; more generally, $\|D\|_p$ is the $\min/\max$ of $\|f\|_p$ for fixed $p\in[0,2]/[2,\infty]$. In the paper this is proved for the special case where $f(z)=1\pm z\pm z\pm z^2\pm\dots\pm z^{N-1}$ and $p\in[0,4]$, by first proving stronger results for the eigenvalues of finite sections of the Toeplitz matrices of $|D|^2$ and $|f|^2$, in particular, for their Schatten $p$-norms. Several conjectures are also stated to the effect that these stronger results should be true for the general case of $f$. The approach is motivated by the uncertainty principle and two theorems of Sze̋go.