I'm looking for a reference that basically states that the Krull dimension of an algebraic set in $\mathbb{R}$ equals the Minkowski dimension of the set.
It is common knowledge that the Krull dimension equals "The maximal dimension of the tangent vector spaces at the nonsingular points of V", however I couldn't even find reference to this this fact.
Also, to avoid singularities I prefer a statement that links the Minkowski dimension to the Krull dimension.