Same problem - I purchased one for my daughter and 6 weeks later the screen is white. I've tried restoring to factory settings with no result. I am now going to see if I can resolve the problem by letting the battery go completely flat before recharging - if this fails the Ipod will be binned and not replaced till Apple releases a reliable product.
"repaired" I hope you mean the kind of repair that iTunes provides?
I strongly suggest using this: http://www.apple.com/support/ipodnano/contact/
They'll call you and you'll probably need a repair or replacement this doesn't look like something that can be fixed by yourself.