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Infinite staircases in the symplectic embedding problem for four-dimensional ellipsoids into polydisks

We study the symplectic embedding capacity function $C_β$ for ellipsoids $E(1,α)\subset R^4$ into dilates of polydisks $P(1,β)$ as both $α$ and $β$ vary through $[1,\infty)$. For $β=1$ Frenkel and Mueller showed that $C_β$ has an infinite staircase accumulating at $α=3+2\sqrt{2}$, while for integer $β\geq 2$ Cristofaro-Gardiner, Frenkel, and Schlenk found that no infinite staircase arises. We show that, for arbitrary $β\in (1,\infty)$, the restriction of $C_β$ to $[1,3+2\sqrt{2}]$ is determined entirely by the obstructions from Frenkel and Mueller's work, leading $C_β$ on this interval to have a finite staircase with the number of steps tending to $\infty$ as $β\to 1$. On the other hand, in contrast to the case of integer $β$, for a certain doubly-indexed sequence of irrational numbers $L_{n,k}$ we find that $C_{L_{n,k}}$ has an infinite staircase; these $L_{n,k}$ include both numbers that are arbitrarily large and numbers that are arbitrarily close to $1$, with the corresponding accumulation points respectively arbitrarily large and arbitrarily close to $3+2\sqrt{2}$.