"I offer you my house."
Translation:Jeg byr deg huset mitt.
Somebody (I don't remember who :( ) wrote this in the discussion of a similar sentence. There the question was the same, just the other way round: "Why is 'på' necessary? That sentece was something like "What does she offer him?"
If you can somehow look up the sentences that recenty came up for me, you might find it in the comments.