Non-existence of solutions for the periodic cubic NLS below $L^2$

We prove non-existence of solutions for the cubic nonlinear Schr"odinger equation (NLS) on the circle if initial data belong to $H^s($\mathbb{T}$) \setminus L^2($\mathbb{T}$)$forsome$s \in (-\frac18, 0)$. The proof is based on establishing an a priori bound on solutions to a renormalized cubic NLS in negative Sobolev spaces via the short time Fourier restriction norm method.