Suppose I have 65 people that register on January 1, 2012.
I want to find out how many of those 65 people returned to the site that same week. (More generally, if n people signup on date A, I want to be able to find out how many of those n people return in a given date range.)
Is there a way to do this using Google Analytics? If so, how? I am currently getting the user's username for each page hit.
If you only need to track people who sign in then you don't need to get very fancy. You can copy the relevant user attributes, such as sign up date, from your DB to GA using events or session level custom variables.
But if you want to track everyone, including those who don't sign up, then you'll need to use visitor level custom variables (GA cookies).
I explain how to set this up in detail in this post so I'll just highlight the key points here:
Once you've put in the work your new visitors will be stamped by their first visit date and nicely fall into each daily or weekly retention bucket. This is what it might look like if you were tracking weekly retention, for example:

This is not a full solution, but here are some points on how I would approach this problem with the help of Google Analytics:
You have to make sure that you somehow store the registration date of each user, either in your database or in a cookie. Then have a look at Google Analytics Event Tracking. You could for example set up a new category based on the registration date. On every page load in your page, you then have to set up this event tracking call, for example like:
_trackEvent("returns", "2012-01-01", "UserId:123123123")
This way you will receive all page views for users that registered on that particular date. To add a date range in this, you have to make sure that these events only get fired for the number of dates after the signup (e.g. 7 days).
After your date range, you will be able to see how many page views and how many users returned - you even know which users came back.
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