IMAP "push" usually means the IDLE extension, where a client holds a connection open and waits for the server to report that something has happened. This works ok, but isn't great on mobile because holding a TCP connection open is usually difficult on flaky networks and consumes battery.

iOS Mail doesn't implement it, instead doing a poll every 15 minutes. However it also implements a separate push system which allows true push if the server supports it. The details on how to support this aren't public information.

We talked to Apple, and they were kind enough to give us access to this system. We implemented it on our side, and now when an email arrives at FastMail we can immediately signal to Mail.app that there's new mail available.