  We prove that the the mixing time of the Glauber dynamics for sampling
independent sets on $n$-vertex $k$-uniform hypergraphs is $O(n\log n)$ when the
maximum degree $\Delta$ satisfies $\Delta \leq c 2^{k/2}$, improving on the
previous bound [BDK06] of $\Delta \leq k-2$. This result brings the algorithmic
bound to within a constant factor of the hardness bound of [BGG+16] which
showed that it is NP-hard to approximately count independent sets on
hypergraphs when $\Delta \geq 5 \cdot 2^{k/2}$.
