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.
Categories: math.CA math.AP