\frac12}|{\widehat}{n_0}(k)| < \infty$$ almost surely. Hence the conditions of Theorem \[mainThm\] are met for $r = \beta - \frac12$.

As a corollary of the proof of the main theorem, we also have the following Sobolev space global well-posedness result, which matches the sharp local theory up to the endpoint.

The one-dimensional Zakharov system is globally well-posed for initial $(u_0, n_0^\pm) \in H^{s}({\mathbb{T}}) \times L^2({\mathbb{T}})$ for any $s > \frac12$.

We will work with the Besov-type Fourier restriction norms defined by $$\begin{aligned}
\| u \|_{X^{s,b, 1}} &= \left\| N^s L^b \| P_{N,L}u\|_{L^2_xL^2_t} \right\|_{\ell^2_N \ell^1_L}, \\
\| n \|_{Y^{s,b, 1}_\pm } &= \left\| N^s L^b \| Q^\pm _{N,L}n\|_{L^2_xL^2_t} \right\|_{\ell^2_N \ell^1_L}, \end{aligned}$$ where $N\gtrsim 1$ and $L \gtrsim 1$ are dyadic and the frequency restriction operators are $$\begin{aligned}
    {\widehat}{P_{N,L}u}(k, \tau) &:= \chi_{|k| \approx N} \chi_{|\tau - k^2| \approx L } {\widehat}{u}(k,\tau) \\
    {\widehat}{Q^\pm_{N,L}n}(k, \tau) &:= \chi_{|k| \approx N} \chi_{|\tau \pm |k|| \approx L } {\widehat}{n}(k,\tau). \end{aligned}$$ We also have time-localized versions of these spaces, denoted $X^{s,b, 1}_\delta$ and $Y^{s,b,1}_{\pm \delta}$, which are defined in the usual fashion. These norms were used for the Zakharov system in [@BHHT]; their properties may be found there. In the following, we often drop the $\pm$ from the 