We have about 80 iPad 4's here, and one just developed this problem. We tried updating to the latest iOS, but it didn't help. I followed some advice I found somewhere to turn gestures off and on, and all keyboard options off and on, but it didn't help.
It seems to randomly tap in the middle of the screen, not always in exactly the same spot. Sometimes it can go for long periods ok, then it'll start up again. It appears to be less affected in portrait than landscape orientation, but I suspect that's only because in landscape there are no icons right in the middle of the screen.
I have discovered that I can replicate the problem by flexing the iPad a little. If I hold the two long edges and twist each hand in opposite directions so that one top edge goes up and the other top edge goes down, I can make it go crazy. I'm making it flex by only a millimetre or two. Holding the iPad by just one corner can also trigger it, as its weight makes it flex.
I now think something inside it is either cracked, loose or dirty, and is making intermittent contact. I've sent it off to a repairer to have a look.
Hopefully this posting will at least help others to easily demonstrate the problem to Apple staff or repairers. If your iPad's problem can be triggered the same way, a workaround might be to put it in a very rigid cover, e.g. Griffin Survivor. That might at least reduce the problem to a tolerable level.