What is this mysterious substance that keeps appearing on my carpet?

Here's a picture. It looks a lot like pepper grains, except it just keeps coming back! I vacuumed it up yesterday, and 24 hours later, it's back. It vacuums very easily, which makes me think it's on the surface of the carpet. There are no holes or defects in the walls or ceiling.
It's actually really creepy.
I would really appreciate if someone can identify this for me and suggest how I can get rid of this! (sorry it's really bad resolution because I took the picture from my iPhone)
We live in an apartment on the top floor of Los Angeles, no pets, and I'm pretty sure no one is deliberately pouring pepper on the carpet.

Added (1).

here's a picture of the individual grains -- i dabbed at it with carpet cleaner which is why there's a wet spot. They're really easy to pick up, are about 1mm in length.
WHAT IS THIS SORCERY!

Added (2). I popped one with the edge of my fingernail, it's really hard, almost like sand, but with pressure, it fragments into pieces and gives off a distinctive mineral like, musky or spicy smell. I don't think i should do that again, it can't be good.

That's pretty weird! I can't tell you what they are, only what it looks like to me which is from termites. It's got to be coming up from the floor. I would cut the carpet to see. After looking I would sew the carpet back up. My curiosity is outrageous and I would have to find the source.

The answer would be easy if by chance these specks are on the carpet below a plant. They look like the classic droppings (poop) from a variety of grubs that eat indoor plants. Even if there's a plant in the room some distance away, they can go walkabout in the night, explaining why you find them again in 24 hrs. Otherwise, still droppings I'd say, perhaps of a beetle that may come in (even on the top floor) if you have those open 'grate' air vents in your room. Less hocus-pocus and highly likely some small creature poop! Will probably only be able to get rid of it when you find the culprit.

In one of the pictures it looks like a big outside roach or a mouse left their feces.
In the other, it looks like a mouse or something carried something to that spot and ate it.