This fix will only be of use to you if you have just factory-resert your Apple TV, performed a new Sync of your videos, you actually use multiple Apple ID's (i.e. you and your partner have separate accounts), and you are seeing the white screen instead of your video.
NOTE: This problem relates to an issue with the FIRST VERSION of the Apple TV. It does not occur on Apple TV 2.
When you select a video (film or TV program) that you have purchased from the Apple Store (i.e. it includes Apple DRM) the screen fills with white and at the bottom you see the "progress" bar which continues to count up but the picture remains just solid white.
There is no audio.
Your Apple TV has not been authorised to play video's for the specific user account that purchased the video.
Replicating The Problem
I've always though this this should be the most important section when describing any IT-related problem. How do you know you've fixed a problem you can't replicate? Needless to say working out exactly what caused this required quite a lot of time!
First of all you need two Appe ID's both of which have to have purchased a video (of some description) from the iTunes Store. Of course this is fairly easy to setup given the number of "free" videos and email services like GMail out there.
Let's call the accounts  and .  has the free pilot episode of Stargate: Universe while  has the free recap of Lost.
Perform a factory reset of your Apple TV and then perform the initial setup from iTunes when it appears. Use account  to register your Apple TV and then sync both videos.
Attempt to play the video from  (Lost) on the newly refreshed Apple TV. You will see the white screen as despite iTunes allowing you to Sync the video it does not have the keys to actually play it.
Fixing the Problem
Ah yes, the important bit! Apple provide a potential solution to the problem here - IT veterans will recognize it as our perenial favourite the "turning it off and on again" answer to everything.
Assuming Apple's recommendation hasn't worked try the following:
- Go to the "Settings" and then the "General" menus
- Select "iTunes Store"
- Log in with the Apple ID that purchased the video (following the example above this would be account )
- Go back to the main menu and select "Downloads" and then "Check for Downloads"
Now when you try and play the video it will work (as the Apple TV has now been authorised to play videos from this account). You should be able to repeat this with as many additional accounts as necessary.