Some people might argue that "based on" has no subject. But, for the sake of discussion , I'll say that I see the entire clause he set about to hunt down the owner of the ring as the subject of "based on". If we rearrange the sentence as His setting about to find the owner of the ring was based on the few clues the ring provided , like the year 1938 and the initials etched inside the band.
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CliveHi,Perhaps it's a British English / American English difference?CliveThe Cambridge online dictionary seems to support your claim that only "verb+ing" is possible (correct) after "set about" in BrE. So, you could be right about this being yet another AmE /BrE difference.