What is the ruling on taking and using the remnants of corn or barley harvest from government lands that are harvested by machines that leave crop residues?

It is permissible to take what the harvester has left behind and benefit from it if it is not known that they will return for it. What the harvester leaves of crops and what the gleaner leaves of dates, out of disinterest, becomes the property of whoever takes it, whether they are a Muslim or a Dhimmi.