to prove, see for instance [@chnava] and Lemma \[lemma\_ell3/2\]:

\[l:frequency\_comparison\] There exists $C(n,\lambda)>0$ and $r_0(n,\lambda)>0$ such that if $r_x>Cr$ with $r\leq r_0$, then $\inf_{B_r(x)}|\nabla u|^2 > \frac{n}{4r^2}\fint_{\partial B_{2r}(x)} |u-u(x)|^2$. In particular, we have the inclusion $$\begin{aligned}
{\mathcal{C}}_r(u)\subseteq\{x\in B_{1/2}(0):\ r_x\leq Cr\}\, .\end{aligned}$$

The above Lemma allows us in the proof of Theorem \[t:main\_critical\] to prove a volume estimate on the set $\{r_x<r\}$, which will be more natural in the context of the frequency.

Let us now generalize the above construction, as it will play an important role in our covering argument. For $d\in{\mathds{N}}$ a fixed integer let us define the following $d$-critical radius:

Given $x\in B_1(0)$ in its domain, we define the $d$-critical radius $$\begin{gathered}
  r^d_x = \sup{\left\{0\leq s\leq r_0: \forall y\in B_s(x) \text{ we have that } N(y,s)< d+\epsilon_0\right\}} \, .
 \end{gathered}$$

Though not supremely important at this stage, the constant $\epsilon_0(n,\lambda)>0$ is chosen from Corollary \[c:cone\_splitting\] by setting $\tau=10^{-6}$, and the radius $r_0(n,\lambda, d)$ is chosen from Theorem \[t:eff\_tan\_con\_uniq\].

For the sake of the outline all that is important is that the constants in the above definition are chosen small enough that one has the effective cone splitting of Corollary \[c:cone\_splitting\].

Outline of Proof {#ss:proof_outline}
----------------

The proof comes 