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 ∈ (0,1] satisfying r ≤ s ≤ (t,u) ≤ (t,u) ≤ v ≤ 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.
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