After upgrading to Mavericks, I see that the Mail app has been upgraded to version 7.0, and that the setting for "Check for new messages" has been changed from my previous setting (every minute) to a new "Automatically" option.
According to the help screen, this means that "Mail varies how often it gets messages, based on whether your Mac is plugged into an electrical outlet". My Mac is plugged in, and I've been keeping an eye on the Activity window, and I've seen intervals of 20 minutes, 7 minutes and 9 minutes between checking for mail. It is, of course, possible that I missed an update - I haven't been obsessively staring the Activity window.
Does anybody know what the intended behavior of the app is, when the "Automatically" option is selected? If not, is there a way to increase the log level to try to work out what the app is actually doing? I've looked in the Console and there appear to be no log events associated with the "Fetching new mail" activity I see in the activity window.