The Hausdorff dimension 
of a subset 
of a metric space 
is the infimum of 
such that the 
-dimensional Hausdorff measure
of 
is 0 (which need not be an integer).
Informally, self-similar objects with parameters 
and 
are described by a power law such as
 |
where
 |
is the similarity dimension of the scaling law (for 
).
In many cases, the Hausdorff dimension correctly describes the correction term for a resonator with fractal perimeter in Lorentz's conjecture. However, in general, the proper dimension to use turns out to be the Minkowski-Bouligand dimension (Schroeder 1991).
Wang (2026) proved, assuming the axiom of constructibility 
for the constructible
universe, that proper subfields of 
that are coanalytic sets
occur with every Hausdorff dimension ![alpha in [0,1]](/images/equations/HausdorffDimension/Inline12.svg)
. The subfields form
a strictly increasing family in 
. For 
, the resulting subfields
are not Borel sets. The construction relates to Erdős
problem 1154 but does not settle
its ZFC formulation. Wang (2026) reports Danus assistance with the proof.
