On certain quantifications of Gromov's nonsqueezing theorem

Let R > 1 and let B be the Euclidean 4-ball of radius R with a closed subset E removed. Suppose that B embeds symplectically into the unit cylinder D-2 similar to R-2. By Gromov's nonsqueezing theorem, E must be nonempty. We prove that the Minkowski dimension of E is at least 2, and we exhibit an explicit example showing that this result is optimal at least for R similar to root 2. In the appendix by Joe Brendel, it is shown that the lower bound is optimal for R <= root 3. We also discuss the minimum volume of E in the case that the symplectic embedding extends, with bounded Lipschitz constant, to the entire ball.