Interpretation of The Gray Havens

[Lotrelf]

Originally Posted by Gwindor

Bilbo and Frodo never actually went all the way to Valinor, they went to the island just off of the main continent named Tol Eresea.

I have wondered about something here. Bilbo, Gimli and Sam, all being mortals are allowed to sail across the Sea without any Jewel like Frodo. What was the need of that Jewel that Arwen gave him? He could have gone there without the jewel too. Does that indicate that Frodo, among all the Mortals, is allowed to go to Valinor too?
Was there any other importance it, Arwen's gem's, or just a "ticket" for Frodo, while others went without any ticket?