The difference between median market rate rents and median stabilized rents are quite small outside of Manhattan south of 96th Street.[1] Many neighborhoods have large number of nominally rent stabilized being offered for preferential rates (i.e. below the legally allowed maximum rent.)

Also, Brooklyn, Queens and the Bronx have plenty of nice, and dense, housing. You should try leaving Downtown Manhattan by Disney World (tm) every once and a while.

[1]http://www.nytimes.com/2012/07/08/realestate/rent-stabilized...