A note on global existence for the Zakharov system on <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{T} $\end{document}</tex-math></inline-formula>

We show that the one-dimensional periodic Zakharov system is globally well-posed in a class of low-regularity Fourier-Lebesgue spaces. The result is obtained by combining the I-method with Bourgain's high-low decomposition method. As a corollary, we obtain probabilistic global existence results in $ L^2 $-based Sobolev spaces. We also obtain global well-posedness in $ H^{\frac12+} \times L^2 $, which is sharp (up to endpoints) in the class of $ L^2 $-based Sobolev spaces.