Coincidence and noncoincidence of dimensions in compact subsets of $[0,1]$

Abstract

We show that given any six numbers $r,s,t,u,v,w\in (0,1]$ satisfying $r\le s\le \min (t,u)\le \max (t,u)\le v\le w$, it is possible to construct a compact subset of $[0,1]$ with Hausdorff dimension equal to $r$, lower modified box dimension equal to $s$, packing dimension equal to $t$, lower box dimension equal to $u$, upper box dimension equal to $v$ and Assouad dimension equal to $w$. Moreover, the set constructed is an $r$-Hausdorff set and a $t$-packing set.

18 pages