Directional maximal operators in the plane

What the math gives to ML

The paper develops a multiscale description of direction sets using recursively nested gap intervals and associated M-adic trees. Its transferable asset is the finite-splitting invariant: a direction family may contain many elements while requiring only a bounded number of genuine branch decisions along any scale path. This suggests replacing dense directional attention or mixture-of-experts routing with a learned hierarchical direction tree having bounded branching and adaptive depth. The most direct target is a vision module that aggregates features along multiple image directions while sharing computation across directions that remain in the same angular interval.