How to exploit knowledge of the sampling distribution for better generalization bounds?
For specific distribution over $X$ uniform convergence bounds scale linearly with the logarithm of the size of the (smallest) $\epsilon$-cover of the concept class $C$ (the cover can use any functions over $X$). This is proved in "Learnability with respect to fixed distributions" paper of Benedek and Itai. As a result sample complexity of PAC learning scales linearly with this parameter.