Welcome to Fluther.
No one mentioned little children, dirty shoes or dogs.
No one seems to have also considered that the “little piles of sand” could very well be “little piles of sawdust” ... from the ceiling. In older homes especially “powder post beetles” bore into ceiling beams and joists, leaving small piles of sawdust (a charitable way of saying “powder post beetle excrement”) on the floor underneath where the work is actually going on. It can be a severe problem – especially when you don’t know that you have them until you see the piles on the floor.
I’d suggest calling an exterminator to see if you have a bigger problem than you think. A good one will tell you exactly what you’re dealing with.