Before revision:  In particular, when specialized to the \varepsilon-LDP model, our protocol achieves an error of k/(e^{\Theta(\varepsilon)}-1) for all \varepsilon, while the previous protocol (Chen et al., 2020) has error O(\sqrt{k/\min}  \{ \varepsilon, \sqrt{\varepsilon\}) } {\varepsilon} \}) bits of communication and O(k \log u) bits of public randomness, where u is the size of the domain} .
After revision:  In particular, when specialized to the \varepsilon-LDP model, our protocol achieves an error of k/(e^{\Theta(\varepsilon)}-1) for all \varepsilon, while the previous protocol (Chen et al., 2020) has error O(\sqrt{k/\min} using O(k\max \{ \varepsilon, \sqrt{\varepsilon\}) } {\varepsilon} \}) bits of communication and O(k \log u) bits of public randomness, where u is the size of the domain} .