All nations (greeks,slavs,armenians) living in borders East Roman Empire; with speech greek are romans by Justinian law from 6.c (like U.S. citizenship), who fink about other nations outside empire like barbarians (like orginal romans). Im not lawer.., but i fink the Byzantine (15/16 .c term.) is true Roman Empire state at least to 4.crusade..,
Francia - first reborn East Roman Empire, HRE - secend reborn Roman Empire, Sejluk Sultanate, Ottoman Empire and Russian Tsardom are all usupers of Roman title. some by religious reasons, some by land, some by political, some by family conections.
Charmagne is great person.., after 200 years of chaos he claim title Emperor. All this discusion about claims is discusion about who have greater peni..
Remember this was dark age.., era of viking and power of the stronger.., if Byz dont agreed and dont accept the new reborn Roman Empire on east, they should do something about it.., something like Justinian..