Tom, Mr.Sloane, and a young lady visit Gatsby and the lady invites Gatsby to come to dinner with them.What does Gatsby's response tell us about his...

I think that this scene is meant in part to show that Gatsby is not very socially sensitive.  Tom points out, once Gatsby is not in the scene, that the invitation was not really sincere.  However, Gatsby does not seem to realize this.

However, I think the scene has more to do with Gatsby's obsession with Daisy.  Gatsby has been hoping, of course, to find a way to casually meet up with Daisy again.  He has been hoping to show her that he has become rich and worthy of her.  Therefore, he will do anything (even if it is expensive or socially awkward) to try to meet her again.