The Ring and Its GPS Tracker

[Inziladun]

I wondered about that morm. However, the Tale of Years states that in TA 1050:

About this time a shadow falls on Greenwood, and men begin to call it Mirkwood.

Then, in 2470:

About this time Sméagol-Gollum hides in the Misty Mountains.

So that's 1,420 years of rebuilding from Sauron.

In 2845 Sauron was aware enough to know who had the last of the Seven, at least, and was powerful enough to bring Thráin II to Dol Guldur to obtain it.

In 2941 Bilbo found the Ring, and only 10 years later, in 2951:

Sauron declares himself openly and gathers power in Mordor.

So there's a 10 year gap between Gollum losing the Ring and Sauron becoming confident enough to abandon secrecy and proclaim to all that he was back.

Now there were 96 years from the time Sauron figured out who had one of the Seven, to the time Gollum lost The One, and he'd been using it pretty regularly, at least to get food. It just seems to me that if Sauron could have had the awareness he obviously had of Thráin's ring, and the ability to get it from him, he should have had some inkling at least that The One still existed, and had not been destroyed.