Nodal intersections and geometric control

We prove that the number of nodal points on an $\mathcal{S}$-good real analytic curve $\mathcal{C}$ a sequence $\mathcal{S}$ Laplace eigenfunctions $\varphi_j$ eigenvalue $-\lambda^2_j$ Riemannian manifold $(M, g)$ is bounded above by $A_{g , \mathcal{C}} \lambda_j$. Moreover, we codimension-two Hausdorff measure $\mathcal{H}^{m-2} (\mathcal{N}_{\varphi \lambda} \cap H)$ intersections with connected, irreducible hypersurface $H \subset M$ $\leq A_{g, H} The $\mathcal{S}$-goodness condition normalized logarithms $\frac{1}{\lambda_j} \operatorname{log} {\lvert \varphi_j \rvert}^2$ does not tend to $-\infty$ uniformly $\mathcal{C}$, resp. $H$. further show satisfying geometric control for density one subsequence eigenfunctions. This gives partial answer question Bourgain–Rudnick about hypersurfaces which can vanish. characterizes positive vanish or just have $L^2$ norms tending zero.