See the arguments to the List Stories endpoint:
http://apidoc.digg.com/ListStories#Arguments
In particular, min_promote_date.
Here's an example, the stories promoted in the 24 hours before I
wrote this email, more or less:
What Jeff said.
If you want the stories with most diggs to appear at the "top" of the
list, you can add the sort arg:
Otherwise, by default the stories are sorted latest promotion first.
Note also that by default the API returns only the first 10 of the
selected stories. Right now, the above URL selects 85 stories. To
see more than the first 10, use the offset and count args.