I don't think keeping sockets open waiting for incoming data have big impact on battery usage because there is no data transmission at that moment so radio shouldn't consume much energy in stand-by mode.

I use K9-Mail app for email working 24h a day, it has multiple accounts on different IMAP4 servers. You know, IMAP requires one keep-alive socket per subscribed folder and I have no problem with battery usage.

I don’t think IMAP is a good example here. Your email client will try both subscribing and classic polling. Subscribing is not a MUST. And from the point of view of end user, the difference when polling in a classical way is simply that the user will be notified later, so it’s hard to tell what the client really did in the background.

> way is simply that the user will be notified later, so it’s hard to tell what the client really did in the background.

I receive emails instantly. There is polling option in settings, I've disabled it.