Regularity, quantitative deviation, and non-rigidity of a lacunary skew product
arXiv:2608.25821
2026
Architecture
2 ideas extracted · analyzed Aug 29, 2026
What the math gives to ML
The paper gives a deterministic multiscale Fourier construction whose frequencies are continued-fraction denominators of an irrational rotation. The resulting series is Holder continuous at every exponent below one but is not Lipschitz, creating a controlled middle ground between smooth positional features and unconstrained high-frequency encodings. The most promising transfer is a compact lacunary positional encoding for neural fields, long-context models, or dynamical predictors. Its skew-product recurrence also suggests a parameter-free state module with bounded phase dynamics and an explicitly accumulated history signal.
Ideas from this paper
Unverified
2026
Replace random Fourier features or a dense sinusoidal positional encoding with a compact bank whose frequencies are the continued-fraction denominators of an irrational number. Inverse-frequency amplitudes provide multiscale structure with a controlled sub-Lipschitz regularity profile, while lacunarity reduces the number of frequencies needed to represent oscillatory structure.
Useful6/10
Difficulty3/10
Novelty5/10
Unverified
2026
Use the paper's skew product as a parameter-free recurrent state: one phase rotates by an irrational increment and a second state accumulates a lacunary Fourier readout of that phase. This supplies deterministic long-range memory with only scalar updates, avoiding a learned recurrent transition matrix and its potentially unstable spectrum.
Useful5/10
Difficulty5/10
Novelty6/10