The Optoma UHD51A is really being well reviewed and is one of the top-picks right now for home theater use. It should be noted that it doesn't have very good input lag measurements, so any serious gaming on it will not bode well for anyone, but for movie viewing and casual gaming it will be fun to use.
These projectors display 8 million pixels discretely. I could get into the details of how they accomplish this, but for the money, let's just say it's been debated, yet the final results are really the proof. They are much more accurate than 1080p projectors, and while not as sharp as the $5,000 Sony native 4K projector, they are very close.
But, instead of just recommending a projector and walking away, why not have a discussion about your room, your use, what you are hoping to accomplish with your setup, and what projector is actually appropriate.
Room size? (l/w/h)
Viewing distance (couch to screen)?
Room paint color (ceiling, walls, carpet)?
Ambient light control (dimmers/windows/ etc.)?
Viewing habits (gaming, movies, sports, general TV)?
Amount of use expected daily?
Sound system that will be paired with projector?
Ability to place projector in room (ceiling/table/distance)?