Quantitative Fourier decay for Patterson-Sullivan measures of dimension larger than $1/2$

What the math gives to ML

The paper provides a constructive quantitative anti-concentration mechanism: when a Patterson–Sullivan measure has dimension \(\delta>1/2\), its characteristic function decays polynomially with explicit exponent \(\alpha(\delta)=\delta(2\delta-1)/((2\delta+1)(3-\delta))\). The proof obtains this from oscillatory-integral cancellation, hyperbolic non-concentration, and a dual transpose-group argument rather than black-box flattening. A transferable neural-network analogue is to regularize normalized one-dimensional projections of embeddings or latent codes so their empirical characteristic functions obey a calibrated power-law envelope, suppressing spectral concentration and potentially reducing collapsed or highly periodic representations. The sharp, falsifiable signature is a change from negligible guaranteed decay below effective dimension \(1/2\) to positive log-log Fourier-decay slope above it.