Nonrigidity of the trigonometric system in $L_p$ for $1\le p<2$

We prove that the trigonometric system ${\mathcal{E}}_N=\{e_k{\}}_{\mathrm{\mid }k\mathrm{\mid }\le N}$ is not rigid in $L_p(\mathbb{T})$ for $1\le p<2$. That is, as $N\to \mathrm{\infty }$, all its elements are approximated with error $o(1)$ by a subspace of dimension $o(N)$.