Quantitative Uniqueness and Rough Damping on $\mathbb T^2$

What the math gives to ML

The paper develops explicit geometric decompositions of thin Fourier annuli at the curvature scale R^{-1/2}, together with discrete bounds on overlaps and difference multiplicities. Its polygonal-boundary result shows that frequency pairs from nonparallel sides have uniformly bounded multiplicity for every fixed difference vector. This suggests a structured Fourier neural layer whose learnable frequency supports are partitioned into transverse sectors or polygonal patches, limiting many-to-one collisions in spectral convolutions. The most practical transfer is an anti-aliasing and variance-control mechanism for Fourier layers and neural operators, rather than a direct use of the PDE observability theorem.