We consider linear combinations of eigenfunctions of the Laplace-Beltrami operator on a compact Riemannian manifold $(M,g)$ and investigate a density property of their zero sets. More precisely, let $f=\sum_{ k=1}^m a_k \phi_{ \lambda_{ j_k}}$, where $-\Delta_g\phi_{ \lambda}=\lambda\phi_{ \lambda}$. Denoting by $Z_f$ the zero-set of $f$, we show that for any $x\in M$, $dist(x,Z_f)\leq C(m)\lambda_{ j_1}^{ -1/2}$. The proof is based on a new integral Harnack-type estimate for positive solutions of higher order elliptic PDEs.
Categories: math.AP math.SP