To anyone who claims it.
I mean, seriously, what are you going to do? Sue the claimants you don’t like in the Hague?
It belongs to the Greeks, who continue to call the City Constantinople and the empire Byzantine, Andrew Baird nothwithstanding, whose literary and church culture is suffused with Byzantium, and whose language if not range of ethnicities came to coincide with Byzantium’s.
It belongs to the Turks, who see the remnants of Byzantium all around them in İstanbul, and whose Ottoman forebears assumed both the mantle and many of the institutions of the Empire of Rum.
It belongs to the Bulgarians, at least in part, because their Orthodoxy was, for the better part of a millennium, Tsarigrad’s—even if Krum did drink wine out of Nicephorus’ skull, and Basil made has name as the Bulgar-Slayer.
It definitely belongs to the Russians, who saw themselves as the Third Rome for centuries.
I’m reminded of one of the many pointless disputes on who claims whom as theirs in Greek history. The target of the dispute was the military feats of the Souliotes: Orthodox by creed, Albanian by language. Who got to claim them?
A Greek author could be expected to say “Obviously us”, but the Greek author in question (who was an Arvanite) had exactly the right answer, even if it was intended as a dodge.
Their legacy was for the whole world.