Topological dimensions and multifractal Renyi dimensions of Polish spaces: a multifractal Szpilrajn type theorem

In this paper we consider the relationship between the topological dimension dim(T)dX and the lower and upper q-Renyi dimensions (D) under bar (q)(X) and (D) over bar (q)(X) of a Polish space X for q is an element of [1, infinity]. Let dim(H) and dim(P) denote the Hausdorff dimension and the packing dimension, respectively. We prove that

dim(H)(X) <= (D) under bar (infinity)(X) <= (D) under bar (q)(X) <= (D) under bar (1)(X), (D) over bar (infinity)(X) <= (D) over bar (q)(X) <= (D) over bar (1)(X) <= dim(P) (X),

for all analytic metric spaces X (whose upper box dimension is finite) and all q is an element of (1, infinity); of course, trivially, (D) under bar (q)(X) <= (D) over bar (q)(X) for all q is an element of [1, infinity]. As a corollary to this we obtain the following result relating the topological dimension and the lower and upper q-Renyi dimensions:

dim(T)(X) = inf(X similar to Y) (D) under bar (q)(Y),

dim(T)(X) = inf(X similar to Y) (D) over bar (q)(Y),

for all Polish spaces X and all q is an element of [1, infinity]; in (1) and (2) we have used the following notation, namely, for two metric spaces X and Y, we write X similar to Y if and only if X is homeomorphic to Y. Equality (1) has recently been proved for q = infinity by Myjak et al.