  Let $(M,g)$ be a $n-$dimensional, compact Riemannian manifold. We define the
frequency scale $\lambda$ of a function $f \in C^{0}(M)$ as the largest number
such that $\left\langle f, \phi_k \right\rangle =0$ for all Laplacian
eigenfunctions with eigenvalue $\lambda_k \leq \lambda$. If $\lambda$ is large,
then the function $f$ has to vanish on a large set $$ \mathcal{H}^{n-1}
\left\{x:f(x) =0\right\} \gtrsim_{} \left( \frac{
\|f\|_{L^1}}{\|f\|_{L^{\infty}}} \right)^{2 - \frac{1}{n}} \frac{
\sqrt{\lambda}}{(\log{\lambda})^{n/2}}.$$ Trigonometric functions on the flat
torus $\mathbb{T}^d$ show that the result is sharp up to a logarithm if
$\|f\|_{L^1} \sim \|f\|_{L^{\infty}}$. We also obtain a stronger result
conditioned on the geometric regularity of $\left\{x:f(x) = 0\right\}$. This
may be understood as a very general higher-dimensional extension of the Sturm
oscillation theorem.
