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"Anon" is completely right. You *can* keep the messages on the server (for a limited time or even infinitely) and, at the same time, NOT download them with your e-mail client over and over again. Here, Boxer seems to have an internal database mismanagement.
Actually, the POP3 protocol does not have any functionality for telling the server that one specific e-mail has been downloaded by one specific client already. So basically this is a client-side issue. Boxer needs to somehow remember whether it has already downloaded certain e-mails or not. Other apps can do that and they DO. Check out "TypeApp" for example. Yet, they these other apps do not have the cool swipe actions that Boxer supports. So: You should work on improving POP. I will then immediately change to Boxer.