The nonsqueezing theorem states that a ball B_R of radius larger than 1 in R^4 (with the standard symplectic structure) cannot be symplectically embedded inside the cylinder D x R^2 where D is the unit 2-disk. I will explain that this result can be quantified as follows: if E is a closed set and the complement of E in B_R symplectically embeds inside D x R^2, then the Minkowski dimension of E is at least 2, and this is optimal in general. The proof uses Gromov’s waist inequality. This is joint work with Kevin Sackel, Umut Varolgunes and Jonathan Zhu.