Critical two-depth Journé packing for bi-parameter and Zygmund rectangles

What the math gives to ML

The paper gives a critical overlap-control principle for incomparable dyadic rectangles. Although depth penalties in either coordinate alone are not summable, complementary exponents s and 1-s jointly yield bounded total mass and exponential integrability of the resulting overlap height. A transferable neural-network use is a congestion-aware router for sparse two-dimensional attention, where multiscale windows are selected subject to a theorem-inspired penalty on deeply embedded overlapping regions. The construction is especially relevant to vision and document transformers with learned rectangular attention patterns.