Compute or bound covering numbers induced by μ

Develop explicit methods to calculate or upper/lower bound the covering numbers N_ρ(ε) and N_ξ(ε) of the index set N under the metrics ρ and ξ induced by a distribution μ on {0,1}^N, directly in terms of μ.

Background

To bound Δn(μ), the paper introduces two μ-induced metrics on the index set N: ξ(i,j)=P(X_i≠X_j) and a subgaussian-increments metric ρ derived from ξ. Several results provide necessary and sufficient conditions for Δ_n(μ) → 0 in terms of the covering numbers Nξ(ε) and N_ρ(ε).

However, translating a concrete μ into quantitative bounds on N_ξ(ε) or N_ρ(ε) remains nontrivial. The open question seeks practical and general techniques to obtain these covering numbers from μ.