I know that Apple is one of the initial investors in ARM Holdings but because most of the world's "smart" cellphones use a variant of the ARM design, you can understand the fear of Apple owning ARM Holdings itself--the possibility that Apple will make the ARM instruction code highly proprietary and/or get access to "secret" CPU instruction codes nobody else knows about.
In short, expect a LOT of scrutiny by the Department of Justice's Antitrust Division in the USA and European Union antitrust regulators on whether Apple can do this--it's possible that the DoJ and/or EU could reject the deal, and Apple may have to spin off its share in ARM Holdings.
I'd expect no more scrutiny than
Oracle buying Sun Microsystems
Microsoft trying to buy Yahoo
Google buying Admob
I swear you guys are manufacturing stuff that does not exist. Other than PA Semi and potentially Intrinisity Apple does not play in this area which means it would be almost impossible to logically accuse Apple of potential wrong doing because this would be such a vertical move.
Most of the fears that i've read hear about shutting off the competition would get laughed out of court if it were ever allowed. The case for presenting Apple as a danger to the smartphone world would be a tough case to win.
Keep in mind these same DOJ and EU buffoons have tried to nail Microsoft for years and have come up with some lame resolutions (Microsoft please leave Internet Explorer off of your computers here ..thanks)
If Apple buys ARM don't expect there to be much of a fight. They have a right to acquire talent and assets just like any other company.