example; $H^t(K)=\infty$, and $H^s(K)=0$ for all $s > t$

Question

Let $n\in \mathbb{N}$, and $H^s$ be the $s$-dimentional Hausdorff measure.

Does there exist $0<t<n$, and $K \subset \mathbb{R}^n$; compact set

such that $H^t(K)=\infty$, and

$H^s(K)=0$ for all $s > t$?