Disc Authorization Failure. We are unable to verify that your disc is a valid copy of The Sims 3. Please contact customer support.

I've also got a Mac, I've only just started playing last week.

I've found that, so far, only the disc starting version of the base game actually allows me to play. I downloaded a patch to patch the launcher up to 1.67 and from there on immediately this error came up. I'm not really sure what I'm supposed to do, but I found uninstalling the game and reinstalling it seems to work (don't worry, the save files are kept)