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

TL;DR

This paper demonstrates the construction of compact subsets of [0,1] with prescribed Hausdorff, box, packing, and Assouad dimensions satisfying certain inequalities, showing the possible coexistence of these dimensions.

Contribution

It provides a method to construct compact sets with specified multiple fractal dimensions, illustrating the relationships and independence among these dimensions.

Findings

Constructed sets with prescribed dimensions satisfying given inequalities

Set is both an r-Hausdorff and t-packing set

Shows coexistence of various fractal dimensions in compact subsets

Abstract

We show that given any six numbers $r,s,t,u,v,w \in (0,1]$ satisfying $r \leq s \leq \min(t,u) \leq \max(t,u) \leq v \leq w$, 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.