I think it depends on the size of the company sometimes, will the offer have to go through HR or a head office, or straight from the person you spoke to?
I got offered my job in the July, waited 3 weeks for the letter of confirmation, then started in beginning of September.
Knowing what I know now, my letter came from the opposite end of the country where I am based and there is all office procedures to take into account.
I agree looking eager and keen is good, but 3 days might just be a bit too soon. Give it maybe a week is my advice.
Good luck!