The Hobbit and The Lord of the Rings take the chronology of Tolkien's Middle-earth only as far as a couple of centuries after the end of The Lord of the Rings. This is the time which the Wise often call "The Dominion of Men":
The Elder Days are gone. The Middle Days are passing. The Younger Days are beginning. The time of the Elves is over, but our time is at hand: the world of Men, which we must rule.
(Saruman, in The Lord of the Rings, Book II, Chapter 2, "The Council of Elrond")
One can guess, or could guess, that at some point during this time Erebor would be again deserted by the Dwarves, and the Arkenstone—if it were not taken by the last to leave the Mountain—would be left to be looted by later "explorers". But we are not told this in any of the published writings. The last we are told about the dwarves of Erebor is that
After the fall of Sauron, Gimli brought south a part of the Dwarf-folk of Erebor, and he became Lord of the Glittering Caves.
It thus appears that the kingdom of Erebor persisted for at least a while after the end of the Third Age; the copy we have of the Red Book is that which was edited by Findegil, scribe of Gondor, in 175 of the Fourth Age, and it apparently contains no reference to the disappearance or decline of Erebor. There is no reference to what might have happened after that.