Translate high-dimensional rectangle discrepancy lower bounds to gamma-VC dimension

Determine how to translate lower bounds for the discrepancy of axis-parallel rectangles in high-dimensional point sets into lower bounds for the gamma-VC dimension of the symmetric class of axis-parallel rectangle classifiers in R^d, accounting for the requirement that every distribution over the point set be gamma-realizable.

Background

The paper relates the gamma-VC dimension of axis-parallel rectangle classifiers to geometric discrepancy. Existing discrepancy lower bounds concern colorings or, naturally, the uniform distribution, whereas gamma-interpolation requires weak realizability for every distribution on the point set, including highly non-uniform distributions. The authors explicitly state that it is unclear how to perform this translation. They later resolve the issue by adapting a discretized version of Roth’s method, so the passage identifies an unresolved question motivating the subsequent argument rather than an open problem left unresolved by the paper.