In previous work [11], the first author and collaborators showed that the leading asymptotics of the embedded contact homology (ECH) spectrum recovers the contact volume. Our main theorem here is a new bound on the sub-leading asymptotics.

ECH (embedded contact homology) capacities give obstructions to symplectically embedding one symplectic four-manifold with boundary into another. We compute the ECH capacities of a large family of symplectic four-manifolds with boundary, called 'concave toric domains'. Examples include the (nondisjoint) union of two ellipsoids in R 4 . We use these calculations to find sharp obstructions to certain symplectic embeddings involving concave toric domains. For example: (1) we calculate the Gromov width of every concave toric domain; (2) we show that many inclusions of an ellipsoid into the union of an ellipsoid and a cylinder are 'optimal'; and (3) we find a sharp obstruction to ball packings into certain unions of an ellipsoid and a cylinder.Contents c k (X, rω) = rc k (X, ω).

We show that every (possibly degenerate) contact form on a closed three-manifold has at least two embedded Reeb orbits. We also show that if there are only finitely many embedded Reeb orbits, then their symplectic actions are not all integer multiples of a single real number; and if there are exactly two embedded Reeb orbits, then the product of their symplectic actions is less than or equal to the contact volume of the manifold. The proofs use a relation between the contact volume and the asymptotics of the amount of symplectic action needed to represent certain classes in embedded contact homology, recently proved by the authors and V. Ramos.

Abstract. In previous work, the second author and Müller 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 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.