If this dimmer switch is on the wall in a house in the usual way then I go along with all the replies above.
However, if you were willing to think laterally and move it (or have a bigger pattress) you could rotate the existing pot with a servo or stepper motor using the Arduino. Always a bit risky and given that you seem a bit new at this game it might not be something you should undertake (now there's a word not to be used in this reply) as you are dealing with mains electricity.
But I mention it only as a bit of lateral thinking and not because it answers your original query.
Just suggestin' 