by working inductively on $d$-critical balls. In this short subsection we will try to outline the general idea of the construction without worrying about precision or technical details.

Let $u$ be a solution of (\[eq\_Lu\]) such that the frequency is bounded by $\Lambda$ as in the statement of Theorem \[t:main\_critical\]. Using the results of Sections \[s:harmonic\] and \[s:general\_elliptic\] it is not hard to see that there exists a constant $C(n,\lambda)$ and an integer $d\leq C\Lambda$ such that for each $x\in B_{1/2}(0)$ we have the $d$-critical radius bound $r^d_x\geq r_0$, where $r_0\geq \epsilon(n,\lambda)^{-d}$. We can cover $B_{1/2}(0)$ by at most an exponential number of such balls, thus there is no harm in estimating each such ball individually and adding up the error. Note that on each such ball that after rescaling $r_x\to 1$ and translating $x\to 0$ we can assume we are working on a ball $B_1(0)$ such that $r^d_0\geq 1$.

Now let us fix $r>0$ and assume $B_1(0)$ is such that $r^d_0\geq 1$ as above. The rough goal is to find a collection of balls $\{B_{r_i}(x_i)\}\subseteq B_1(0)$ with the following properties:

1.  For each $i$ either $r_i=r$ or $r_i = r_{x_i}^{d-1}$.

2.  If $x\not\in \cup B_{r_i}(x_i)$ then $r_x<r$, which is 