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

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