Brjuno condition through best approximations and the linearization problem

What the math gives to ML

The paper provides a constructive multiscale mechanism for analytic linearization: Fourier modes are eliminated sequentially, while modes with small divisors are retained inside shrinking resonance cones. The relevant quantitative object is a Brjuno-type summability budget over successive best approximations or switching times of a diagonal lattice flow; this budget controls the loss of analyticity of the conjugacy. A transferable neural-network analogue is a resonance-aware Fourier or state-space training schedule that preconditions or postpones modes whose phase denominators are small, rather than allowing their updates to destabilize the whole model. This is most plausible for Fourier neural operators, periodic sequence models, and learned dynamical systems with an identifiable transport frequency.