On certain quantifications of Gromov's non-squeezing theorem

TL;DR

This paper investigates the minimal size and dimension of subsets that obstruct symplectic embeddings of Euclidean balls into cylinders, establishing lower bounds on Minkowski dimension and providing explicit examples for optimality.

Contribution

It proves that the Minkowski dimension of the obstruction set must be at least 2, and constructs examples showing this bound is sharp for certain radii, advancing understanding of symplectic embedding obstructions.

Findings

Minkowski dimension of the obstruction set is at least 2.

Explicit examples demonstrate the optimality of the dimension bound for R ≤ √2.

Lower bound on the obstruction set's dimension is also optimal for R < √3.

Abstract

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 $\mathbb{D}^2 \times \mathbb{R}^2$. By Gromov's non-squeezing theorem, ${E}$ must be non-empty. 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 $R \leq \sqrt{2}$. In an appendix by Jo\'e Brendel, it is shown that the lower bound is optimal for $R < \sqrt{3}$. We also discuss the minimum volume of ${E}$ in the case that the symplectic embedding extends, with bounded Lipschitz constant, to the entire ball.