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.
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