Some people advocate filling the holes with water or other things to stop the digging. In my opinion, a better policy would be to find an out of the way area for your dog to dig, and not to worry about it. Cairn Terriers can be quite verbal. A Cairn left on a line in the backyard may well become a problem barker, due to loneliness. He will also get eaten by the big dogs in the neighborhood, because he doesn't know when to back down.
Cairns are very territorial.
This is good for security purposes, but can become problematic if precautions aren't taken. Letting a Cairn run loose can give him the impression that a whole neighborhood belongs to him, and woe unto any strange dog trespassing on his land. As mentioned above, Cairns are chewers. They have also been known to dig and climb. Those are the bad points.