Quantitative time-averaged spherical means along equidistributed spirals: Diophantine rates and limits of uniformity

What the math gives to ML

The paper provides a constructive deterministic sampling mechanism: a continuous Kronecker flow on a torus, mapped measure-preservingly to a sphere, achieves uniform time-averaging with an explicit rate controlled by observable smoothness and the Diophantine quality of the frequency vector. This can transfer to neural-network training as a deterministic replacement for random spherical perturbations, augmentation directions, or zeroth-order gradient probes, with a measurable discrepancy bound rather than an informal claim of coverage. The key engineering variable is the frequency vector: poorly approximable frequencies avoid long resonant gaps, while the theorem predicts how empirical averages should decay with the number of probes. The paper also shows that no fixed frequency gives a universal convergence rate for every smooth observable, so experiments must monitor observable-specific discrepancy.