Exponential convergence can happen in weighted Birkhoff averages via quasi-periodicity with arbitrary nonresonance and low regularity

What the math gives to ML

The paper gives a constructive acceleration mechanism for weighted Birkhoff averages along any nonresonant torus translation, even when the observable has only absolutely summable Fourier coefficients and may fail every positive Holder condition. The transferable asset is frequency-aware signed windows whose Fourier response suppresses the small-denominator errors generated by quasi-periodic trajectories. In neural networks, this suggests a quasi-periodic phase module and checkpoint or output averaging scheme for long-horizon state-space models and recurrent networks, with windows designed from the observed phase frequencies rather than using uniform averaging. The key falsifiable signature is stretched-exponential error decay in the averaging horizon, with log-error proportional to negative N raised to a chosen exponent sigma.