  We study the query complexity of finding a Tarski fixed point over the
$k$-dimensional grid $\{1,\ldots,n\}^k$. Improving on the previous best upper
bound of $\smash{O(\log^{\lceil 2k/3\rceil} n)}$ [FPS20], we give a new
algorithm with query complexity $\smash{O(\log^{\lceil (k+1)/2\rceil} n)}$.
This is based on a novel decomposition theorem about a weaker variant of the
Tarski fixed point problem, where the input consists of a monotone function
$f:[n]^k\rightarrow [n]^k$ and a monotone sign function $b:[n]^k\rightarrow
\{-1,0,1\}$ and the goal is to find an $x\in [n]^k$ that satisfies $either$
$f(x)\preceq x$ and $b(x)\le 0$ $or$ $f(x)\succeq x$ and $b(x)\ge 0$.
