Fractal uncertainty principle over $\mathbb{Q}_p$

What the math gives to ML

The paper proves a quantitative uncertainty principle: a function whose Fourier support lies in a porous frequency set cannot concentrate much L2 energy on a simultaneously porous spatial set, with the concentration bounded by a polynomial factor h^beta. This is a structured alternative to generic spectral regularization because it penalizes joint spatial-frequency concentration across reciprocal scales. A practical transfer is a Fourier-domain feature regularizer for convolutional or token-mixing layers, using soft porous masks and penalizing representations that violate the theorem-inspired energy bound. The exact theorem concerns continuous functions and hard support constraints, so the neural version should be treated as a falsifiable surrogate rather than a guaranteed theorem.