Entropy Density of Uniquely Ergodic Measures for Full Shifts over Amenable Residually Finite Groups

What the math gives to ML

The paper provides a constructive block-replacement mechanism for approximating ergodic invariant measures using controlled symbolic blocks. Its transferable machinery is a Følner tiling, separation of block interiors from boundaries, and a quantitative propagation bound showing that local-frequency errors remain small when the boundary-to-volume ratio is small. This can be transferred to sequence-model augmentation and curriculum learning: construct long, globally coherent training contexts by replacing block interiors with samples from a frequency-matched library while preserving boundary regions. The falsifiable prediction is that distortion of local pattern statistics decreases with block size according to the boundary-to-volume ratio, rather than growing with sequence length.