We study the symplectic embedding capacity function
for ellipsoids
into dilates
of polydisks
as both
and
vary
through
. For
, Frenkel and Müller
showed that
has an infinite
staircase accumulating at
,
while for
integer ,
Cristofaro-Gardiner, Frenkel and Schlenk found that no infinite staircase arises. We show that for
arbitrary
, the
restriction of
to
is
determined entirely by the obstructions from Frenkel and Müller’s work, leading
on
this interval to have a finite staircase with the number of steps tending to
as
. On
the other hand, in contrast to the results of Cristofaro-Gardiner, Frenkel
and Schlenk, for a certain doubly indexed sequence of irrational numbers
we find that
has an infinite
staircase; these
include both numbers that are arbitrarily large and numbers that are arbitrarily close to
, with the
corresponding accumulation points respectively arbitrarily large and arbitrarily close
to
.
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