\in H^s({\mathbb{T}})$ and $n_0^\pm \in \mathcal{FL}^{\beta, \infty}$, with $$s> \frac12 \quad \text{ and } \quad \beta >\frac1{2(2-s)}  .$$ Then for any $T >0$, the Zakharov system with initial data $(u_0, n_0^\pm)$ has a solution on the time interval $[0,T]$. Furthermore, the norm of the nonlinear part of the solution grows at most polynomially in time. In particular, $$\begin{aligned}
    max _{0 \leq t \leq T} \Bigl( \| u - e^{it\delta}u_0 \|_{H^s} + \| n^\pm - e^{\pm i t \partial_x} n_0^\pm \|_{L^2} \Bigr) \lesssim C(\|u_0\|_{H^s}, \|n_0^\pm\|_{\mathcal{FL}^{\beta, \infty}}, s, \beta) {\langle}T{\rangle}^{\max\{ \alpha(1-s)\gamma, (\frac12 - \beta) \gamma\} }, \end{aligned}$$ where $$\alpha \in \Biggl( 1 - 2\beta   , \min\Bigl\{\frac{\beta + \frac12 - s }{2(1-s)}, \; \frac{2\beta - s   }{1-s}  \Bigr\} \Biggr)$$ and $$\gamma > \max\Bigr\{ \frac{1}{(2s-1)\alpha}, \frac{1}{\beta + \frac12 - s - 2\alpha(1-s)}, \frac1{\alpha + 2\beta -1   },  \frac{1}{2\beta - s    - \alpha(1-s)} \Bigl\}.$$

It is probable that our methods would also apply to rougher Schrödinger data (that is, to $ u_0 \in \mathcal{FL}^{\alpha, \infty}$, with $\alpha \leq 1$). However, we do not pursue this here.

We note that Bourgain gives a local theory result for wave data in a subset of $H^{-\sigma}$, with $0 < \sigma \ll 1$. We provide an alternate proof of 