I mean successor in two distinct ways here.

1- Did it explicitly claim succession from the Eastern Roman Empire? Did the Sultan of the Ottoman Empire call himself the "Sultan of Rum" or somesuch, like the earlier Turkish state that had emerged following the defeat of the Eastern Romans at Manzikert?

2- How much continuity was there between the Late Roman institutions and administration and the Ottoman one? Did the Ottomans keep Roman bureaucrats? How did they interact with the Orthodox Church?