Don't listen to these twits. It doesn't sound like you're doing anything wrong. Do you feel like she doesn't want you around? If you've gotten no indication that you make her uncomfortable then keep being nice to her. Sometimes even when people act like they couldn't care less they're really just crying out for someone to care and to not give up on them. If you do get the feeling that you make her uncomfortable then of course back away. You know the situation better than anyone here and you could try and explain the subtleties of it till you're blue in the face and someone here would have a problem with what you're doing.

Unless of course you think this girl doesn't want you around I say give her the book yourself.