keep the arguments as non-convoluded as possible we will concentrate on proving Theorem \[t:main\_critical\] in the context where (\[eq\_Lu\]) is critical, as the other results are completely analogous.

The main new ingredient to the proof of Theorem \[t:main\_critical\] is a new covering argument, which itself relies on a new [*effective*]{} tangent map uniqueness statement. In section \[ss:tangent\_maps\] we review the notion of a blow up and discuss the new results in this paper related to them. In section \[ss:critical\_radius\] we discuss the notion of the critical radius. In section \[ss:proof\_outline\] we outline the proof of Theorem \[t:main\_critical\], and in particular the new covering argument.

Generalized Frequency
---------------------

For solutions of (\[eq\_Lu\]) it is more natural and convenient on small scales to work with a generalization of the frequency function (\[d:frequency\]) which takes into account better the coefficients of the equation. Among other things this allows one to preserve the essential property of [*monotonicity*]{} for the frequency. Such a generalized frequency was first introduced in [@galin1; @galin2; @hanlin], and further expanded in [@chnava]. We will follow the mild extensions given in [@chnava], which are discussed in Section \[ss:generalized\_frequency\]. For now, we simply wish to remark that the frequency $N$ in the 