Symplectic embeddings of four-dimensional ellipsoids into integral polydiscs

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.