Why did the Greek language gradually replace Latin in the Eastern Roman Empire? When could this have happened?

Unlike the West, the Roman conquerors did not bring a higher culture to the East than the locals already had. Therefore, Latin there never could supplant the Greek language. With the decline of the state at the end of the 6th century, Greek became ubiquitous. Since the 7th century, emperors have been using Greek nicknames, for example Constantine IV Pogonatus (i.e. the Bearded)

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