For a non-empty compact subset $K\subset \mathbb{R}^d$ and $s\in\mathbb{R}$ denote $H^s_K$ the set of Sobolev-functions of order $s$ which are supported in $K$. Define $$\delta(K)= \inf \{\epsilon\ge0: H^{-\epsilon}_K \neq 0 \} \in [ 0, d/2].$$ When $K$ has positive ($d$-dimensional) Lebesgue measure, then obviously $\delta(K)=0$. Further, if $K$ is a subset of an $m$-dimensional submanifold of $\mathbb{R}^d$ and has positive ($m$-dimensional) Lebesgue measure, then $\delta(K)=(d-m)/2$. This suggests to view the quantity $$ \dim_S K := d - 2 \delta(K)\in [ 0, d] $$ as a dimension of some sort.
Question: Has this notion been studied anywhere, in particular is it known whether $\dim_S$ coincides with some other notion of dimension, e.g. the Hausdorff dimension, especially when taking non-integer values?