Symplectic embeddings of four-dimensional ellipsoids into integral polydiscs

TL;DR

This paper characterizes symplectic embedding functions of four-dimensional ellipsoids into integral polydiscs, revealing a transition from complex number-theoretic obstructions to a regular infinite staircase pattern as the polydisc parameter increases.

Contribution

It extends previous work by determining the embedding function for all integer b ≥ 2, showing how the structure simplifies and converges to a regular pattern as b grows.

Findings

For fixed b, obstructions vanish for large a except volume.

The complex 'Pell stairs' structure simplifies as b increases.

The embedding function converges to a regular infinite staircase pattern.

Abstract

In previous work, the second author and M\"uller determined the function $c(a)$ giving the smallest dilate of the polydisc $P(1,1)$ into which the ellipsoid $E(1,a)$ symplectically embeds. We determine the function of two variables $c_b(a)$ giving the smallest dilate of the polydisc $P(1,b)$ into which the ellipsoid $E(1,a)$ symplectically embeds for all integers $b \geqslant 2$. It is known that for fixed $b$, if $a$ is sufficiently large then all obstructions to the embedding problem vanish except for the volume obstruction. We find that there is another kind of change of structure that appears as one instead increases $b$: the number-theoretic "infinite Pell stairs" from the $b=1$ case almost completely disappears (only two steps remain), but in an appropriately rescaled limit, the function $c_b(a)$ converges as $b$ tends to infinity to a completely regular infinite staircase with steps all of the same height and width.