Ergodic $\times p$-invariant measures on $\mathbb{T}^2$ with no dimension dropping projections

What the math gives to ML

The paper constructs ergodic ×p-invariant measures whose mass in every planar tube of width w is O(w^s), so no line projection loses dimension, even when the projected iterated-function system has exact overlaps. The transferable mechanism is a uniform Frostman/anti-concentration condition over all projection directions, rather than an average-case random-projection guarantee. In neural networks, this can become a representation regularizer that penalizes excessive concentration of hidden embeddings inside thin slabs or intervals after any one-dimensional projection. The main falsifiable prediction is that the worst-direction projected small-ball probability follows a power law with exponent s, while unconstrained representations exhibit a direction and scale range with a smaller exponent.