depending on whether the equation is critical or not. Notice that for every $r>0$ we have that ${\mathcal{CS}}(u)\subseteq {\mathcal{CS}}_r(u)$, and more effectively that points of ${\mathcal{CS}}_r(u)$ are those points which have a definite amount of gradient on a ball of definite size.

#### Background:

To control the critical and singular sets of a solution to (\[eq\_Lu\]) more information about the solution is needed. For instance, one could just take the solution $u=0$, which by all regards is a great solution but there is no control on the critical and singular set. It has been understood for some time that being a constant or close to a constant is all that can really go wrong, and hence what is important is to control how far away $u$ is from a constant solution. The right measurement for this is are the frequencies $$\begin{aligned}
\label{d:frequency}
N^u_{{\mathcal{C}}}(x,r)\equiv \frac{r\int_{B_r(x)} |\nabla u|^2}{\int_{\partial B_r(x)} (u-u(x))^2}\, ,\,\, N^u_{{\mathcal{S}}}(x,r)\equiv \frac{r \int_{B_r(x)} |\nabla u|^2}{\int_{\partial B_r(x)} u^2}\, ,\end{aligned}$$ and their generalizations (see Section \[ss:generalized\_frequency\]), where we denote by $N^u(x,r)$ either $N^u_{{\mathcal{C}}}(x,r)$ or $N^u_{{\mathcal{S}}}(x,r)$, depending on whether (\[eq\_Lu\]) is critical or not, respectively. By unique continuation and the maximum principle, if $u$ is not constant then $N^u$ is well defined for positive 