When you're renting an apartment or house and they run a background check on you as the tenant it can take from 2 to 3 days to hear back from the landlord or housing people.
So once you get the application submitted for the housing then you will wait from 2 to 3 days to hear back because it can take time for the landlord or property manager to run the background check depending on how many more tenants they have to run background checks on.
I once had it take a week before I heard back from the property manager of an apartment when they ran my background.
But it was no problem and usually I would normally hear back within a day or 2.
But if it's been longer than a few days then you might want to contact the landlord or property manager to see what is going on.