A faster algorithm for finding Tarski fixed points

Dang et al. have given an algorithm that can find a Tarski fixed point in a k-dimensional lattice of width n using O(logk n) queries. Multiple authors have conjectured that this algorithm is optimal [Dang et al., Etessami et al.], and indeed this has been proven for two-dimensional instances [Etessami et al.]. We show that these conjectures are false in dimension three or higher by giving an O(log2 n) query algorithm for the three-dimensional Tarski problem. We also give a new decomposition theorem for k-dimensional Tarski problems which, in combination with our new algorithm for three dimensions, gives an O(log2 lceil k/3 rceil n) query algorithm for the k-dimensional problem.