Why Frodo?

I have been reading Christopher Tolkien's books on the creation of the Lord of the Rings and maybe I 'm a bit confused. But I wonder..
Sauron knew (from Gollum) that BILBO or at least some Baggins had found the ring.
In the case he knew that BILBO Baggins was the ringbearer, the ringwraiths should have been informed that he vanished 30 years ago, and search for him (since it was almost impossible for anybody to part from it wllingly)
In the second case (if he knew only the name BAGGINS, anyone under this name was a suspect.
WHY DID THEY INQUIRE ABOUT AND CAME IMMEDIATELY AFTER FRODO?
Or am I confused after reding so many different versions of the story?