Assume that $ E\subset\mathbb{R}^n $ such that $ \mathcal{H}^k(E)>0 $ and for any compact set $ K\subset\mathbb{R}^n $, $ \mathcal{H}^k(E\cap K)<+\infty $, where $ \mathcal{H}^k $ denotes the $ k $-dimensional Hausdorff measure with $ k<n $. Suppose that $ E $ is dilation invariant, i.e. for any $ s>0 $, with $ sE=\{sx:x\in\mathbb{R}^n\} $, $ \mathcal{H}^{m-2}(E\Delta sE)=\mathcal{H}^{m-2}((E\backslash sE)\cup(sE\backslash E))=0 $. I want to ask that if there exists a $ k $-dimsional subsapce $ V $ such that $ \mathcal{H}^k(E\backslash V)=0 $?
Intuitively thinking this may true since if not the dialation invariant condition implies that $ \mathcal{H}^{k+1}(E)>0 $ but I do not know how to show it rigorously. Can you give me some hints or references?