#### Volume 17, issue 2 (2017)

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Symplectic embeddings of four-dimensional ellipsoids into integral polydiscs

### Daniel Cristofaro-Gardiner, David Frenkel and Felix Schlenk

Algebraic & Geometric Topology 17 (2017) 1189–1260
##### Abstract

In previous work, the second author and Müller determined the function $c\left(a\right)$ giving the smallest dilate of the polydisc $P\left(1,1\right)$ into which the ellipsoid $E\left(1,a\right)$ symplectically embeds. We determine the function of two variables ${c}_{b}\left(a\right)$ giving the smallest dilate of the polydisc $P\left(1,b\right)$ into which the ellipsoid $E\left(1,a\right)$ symplectically embeds for all integers $b\ge 2$.

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 ${c}_{b}\left(a\right)$ converges as $b$ tends to infinity to a completely regular infinite staircase with steps all of the same height and width.

##### Keywords
symplectic embeddings, Cremona transform
##### Mathematical Subject Classification 2010
Primary: 53D05
Secondary: 14B05, 32S05
##### Publication
Received: 26 April 2016
Accepted: 12 October 2016
Published: 14 March 2017
##### Authors
 Daniel Cristofaro-Gardiner Mathematics Department Harvard University 1 Oxford Street Cambridge, MA 02138 United States David Frenkel Institut de Mathématiques Université de Neuchâtel Rue Émile-Argand 11 CH-2000 Neuchâtel Switzerland Felix Schlenk Institut de Mathématiques Université de Neuchâtel Rue Émile-Argand 11 CH-2000 Neuchâtel Switzerland