data into high- and low-frequency parts. For the low-frequency part, we apply an I-method local theory, and control growth via an almost-conservation law. The remainder solves a difference equation. We show that the nonlinear part of this difference equation lives in the space where the I-method applies, with small norm. We use this smallness with the almost-conservation law to iterate.

We take initial data of the following form. Fix $u_0 \in H^s({\mathbb{T}})$, for $s \in (\frac12, 1)$, and wave initial data such that: $$\begin{aligned}
\label{eq:wave_data}
n_0^\pm(x) = \sum_{k \in {\mathbb{Z}}/\{0\}} \frac{h_k^\pm}{{\langle}k {\rangle}^\beta} e^{ikx} \qquad \text{with} \qquad \sup_k |h^\pm_k| < \infty.\end{aligned}$$ That is, we take $n_0^\pm$ in the Fourier-Lebesgue space $$\mathcal{FL}^{\beta, \infty} := \Bigl\{ f \in \mathcal{D}'({\mathbb{T}}) \; | \; \sup_k\, {\langle}k {\rangle}^\beta |{\widehat}{f}(k)| < \infty \Bigr\}.$$ We fix $\beta \in (0, \frac12]$, so that $n_0^\pm  \in  H^r({\mathbb{T}})$ for any $ r < \beta -  \frac12$. We require $\beta > s - \frac12$. This corresponds to requiring that $n_0^\pm$ is in a Sobolev space $H^r$ with $s-1 < r$. Note that the case $s-1 = r$ is critical in some sense. When $s-1 > r$, it seems impossible to obtain any well-posedness. Our main result is as follows.

\[mainThm\] Suppose $u_0 