Exact packing measures for collision sets

Identify exact packing measure functions for the collision-time and collision-point sets of independent Markov processes, including the corresponding liminf behavior of the temporal and spatial collision measures at typical collisions.

Background

The paper resolves a broad Hausdorff-measure version of Xiao’s question for collision-time and collision-point sets, but it does not resolve the packing-measure counterpart. The unresolved problem concerns both identifying the correct packing gauges and determining the associated lower fluctuation behavior of the temporal collision measure C_t and spatial collision measure Π_t at typical collisions.

The authors emphasize that packing measures appear substantially more difficult because collision mass would need to accumulate at the target order at every sufficiently small scale, rather than merely along a limsup sequence of scales.