Sets with Different Dimensions in [0, 1]

Abstract

Given $0 < s < u < v < 1$ and $s < t < v$, a Cantor set $Y$ is constructed in $[0, 1]$ with Hausdorff dimension $s$, packing dimension $t$, lower box dimension $u$ and upper box dimension $v$. In the sense that $t$ and $u$ are independent, so are the packing and lower box dimensions. Although $Y = \{0\} \cup \bigcup_{ \ell=0}^{ \infty} Y_\ell$, the lower and upper box dimensions of each $Y_\ell$ are respectively $s$ and $t$.