What are the intellectual achievements of the Byzantines?
This is a great question. I would say that the three greatest intellectual achievements of the Byzantine empire are as follows.
First, the Byzantine empire gave us law codes. For instance, one of the most influential books ever written was Justinian's Digest. It codified Roman law, opinion, and precedents. This was a massive intellectual achievement that shaped Western culture. The importance of this work cannot be overstated.
Second, the Byzantine empire preserved much of Greek and Latin literature. They did not "fall" like the Western Empire. In fact, they continued in their scholarship and passed it on. And when Constantinople fell, many scholars moved to the West and brought their learning.
Third, the Byzantine empire passed on a great theological tradition from Greek speaking church fathers. In this way, they have developed an important theological voice.