$r$. For a fixed solution $u$ of (\[eq\_Lu\]) we then denote by $$\begin{aligned}
\Lambda \equiv N^u(0,1)\, ,\end{aligned}$$ the frequency bound of $u$. The main conjecture in the area goes back to Lin [@lin], which predicts that for some constant $C(n,\lambda)$ we have that $$\begin{aligned}
&{\mathcal{H}}^{n-2}({\mathcal{CS}}(u))< C\Lambda^2\, .\end{aligned}$$ The best that has been proved in the literature at this point goes back to [@hanlin; @hanhardtlin; @HLrank], which proves under the assumption of [*smooth*]{} coefficients there exists constants $C(n,a,b,c,\Lambda)$ such that $$\begin{aligned}
&{\mathcal{H}}^{n-2}({\mathcal{CS}}(u))< C(n,a,b,c,\Lambda)\, .\end{aligned}$$ In particular, there is no effective control on the coefficients at all. If one drops the assumption of smoothness on the coefficients, even if one assumes control over a large number of derivatives but not all, then the situation becomes drastically worse. In this case the best that has been proven is in [@hanlin; @han_sing], where it was shown that Hausdorff dimension satisfies $$\begin{aligned}
\dim{\mathcal{CS}}(u) = n-2\, ,\end{aligned}$$ however it was not even shown that ${\mathcal{H}}^{n-2}({\mathcal{CS}}(u))<\infty$. There is a fundamental reason for this, as the results of [@hanlin],[@lin] rely on an $\epsilon$-regularity theorem which requires additional smoothness as the frequency increases. One of the main goals of this paper is to improve on these estimates by removing 