Oscillatory functions vanish on a large set

Abstract

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.