It's also interesting to note that Mehmed II claimed the title of Caesar after his conquest of Constantinople and that geographically there was a lot of overlap between the greatest extent of Eastern Roman and Ottoman territory. Oh, and not only did a Constantine found the city, but it was the 11th Constantine that died losing it.
Another related historical fact: After Ottomans conquered Constantinople they started to copy Hagia Sophia's architecture to build their mosques. Therefore still the architecture of mosques in old Ottoman lands is similar to Byzantine while the architecture in other muslim world (North Africa, Asia etc.) is pretty different. Ottomans positioned themselves basically as the muslim version of Byzantines. Actually the root of Ottoman Classical Music (now called as Turkish Classical Music) goes to Byzantine music.
https://en.wikipedia.org/wiki/Caesar_(title)#Ottoman_Empire