  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.
