the need for such an $\epsilon$-regularity theorem.

In another direction there are more recent results from [@chnava] that attempt to prove more effective estimates on the critical and singular sets. Namely, even a Hausdorff dimension bound has limited application. In short, the Hausdorff dimension of a set can be small while still being dense. On the other hand, Minkowski estimates bound not only the set in question, but the tubular neighborhood of that set, providing a much more analytically effective notion of [*size*]{}. For example, we recall that the set of rational numbers in ${\mathds{R}}^n$ has Hausdorff dimension $0$ and Minkowski dimension $n$. What is needed for applications to nonlinear equations are control over the critical and singular sets on balls of definite size. That is, it would be better to estimate ${\operatorname{Vol}}(B_r({\mathcal{CS}}(u)))$, and even better to make the statement that if $x\not\in B_r({\mathcal{CS}}(u))$ then the gradient of $u$ on $B_r(x)$ has some definite size. The first results in this direction were proven in [@chnava], where by using the ideas of quantitative stratification it was shown under only Lipschitz coefficients that for every $\epsilon>0$: $$\begin{aligned}
{\operatorname{Vol}}(B_r({\mathcal{CS}}_r(u))) < C(n,\lambda,\epsilon) r^{2-\epsilon}\, .\end{aligned}$$ This gives the much stronger Minkowski estimates on the critical 