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In this paper, we establish local well-posedness for the Zakharov system on$\mathbb{T}^d$, $d\ge3$ in a low regularity setting. Our result improves thework of Kishimoto. Moreover, the result is sharp up to $\varepsilon$-loss ofregularity when $d=3$ and $d\ge5$ as long as one utilizes the iterationargument. We introduce ideas from recent developments of the Fourierrestriction theory. The key element in the proof of our well-posedness resultis a new trilinear discrete Fourier restriction estimate involving paraboloidand cone. We prove this trilinear estimate by improving Bourgain--Demeter'srange of exponent for the linear decoupling inequality for paraboloid under theconstraint that the input space-time function $f$ satisfies ${\rm supp}\,\hat{f} \subset \{ (\xi,\tau) \in \mathbb{R}^{d+1}: 1- \frac1N \le |\xi| \le 1+ \frac1N,\; |\tau - |\xi|^2| \le \frac1{N^{2}} \} $ for large $N\ge1$.

Decoupling inequality for paraboloid under shell type restriction and its application to the periodic Zakharov system