An antichain approach to a conjecture of Zygmund

What the math gives to ML

The paper gives a constructive way to control overlap multiplicity for sparse antichains of dyadic rectangles: the occupancy function h_E has a dimension-reduced exponential moment bound, whereas unrestricted sparse families only obtain an exponential bound for h_E^{1/2}. This is transferable to multiscale vision or 2D-token attention, where nested windows can otherwise cause a small set of tokens to participate in many computation blocks. The most promising adaptation is an antichain-constrained window scheduler or a soft overlap regularizer that limits peak token reuse while preserving multiscale coverage, reducing memory and improving predictable inference cost.