I'm not sure if this directly resulted in the growing gap of wealth between rich and poor. But in the last couple centuries of the Byzantine Empire there was a trend of the aristocracy not paying taxes. Essentially over a couple generations successive Emperors gave concessions to the aristocracy in terms of paying less and less taxes; which greatly hurt the empires coffers. This is in some ways similar to the "don't tax the rich" seen in the US now. Interestingly, the postal service in the Byzantine empire suffered due to the shrinking tax base; which is the same problem plaguing Western countries that have a quickly growing gap between rich and poor (US, Canada, UK). I would imagine, although its not directly stated, that with the aristocracy not paying taxes and the tax burden then more directly falling on the remaining people in society this resulted in a growing income gap. But I highly doubt that the language or concept existed at the time to express that in the way we do now.

The end result. Lack of finances led to an inability fund services and the military. This caused the empire to shrink over time and in ultimately in 1453 when the Ottomans were sacked Constantinople, ending any Byzantine presence in history, they did it with cannons purchased from Genoa -- which the Empire wanted to purchase but could not afford. Its not the only cause of Byzantine decline but probably didn't help their situation.

Resources:

Lost to the West: The forgotten Byzantine Empire the rescued Western Civilization. Lars Brownsworth

Grand Strategy of the Byzantine Empire. Edward N Luttwak