Does anyone have information whether bitcoin would distribute wealth more equally across people than the current wealth distribution?
Or would the distribution be even more unfair?
I guess I'll propose a 2-part answer that depends on what we mean by the 'success of Bitcoin.'
Success could mean (1) Bitcoin becomes the dominant form of currency in the world, or large parts of the world; or (2) Bitcoin becomes a major store of value that is traded at high value in fiat currency terms but is not used much for transactions.
For both scenarios, Bitcoin should have a fundamentally beneficial effect on inequality.
Scenario (1) is frankly not likely -- technical issues aside, as Jamie Dimon said, governments are just not going to let a non-state-controlled money take over. Scenario (2) is looking more and more like where we're headed, where (at least Western) central banks will let Bitcoin go very high and then 'manage' the fiat-BTC exchange rate as a means to support confidence in their fiat money and stabilize the system.
As for the effect on inequality, we first have to understand where the extreme inequality of recent decades, and the inequality of the modern world in general, come from. The current trend comes from central banks being forced to inflate the values of financial assets to make a small no. of people rich, and hoping that their spending will keep the world afloat, or at least that their spending won't collapse and bring the world down with it. This is also an extreme-case microcosm of the modern system in general.
The reason this inequality drives even more inequality is that this is an unstable situation. Spending by the lucky on frivolous luxuries can disappear quickly when the stock market goes down a little, for example. So central banks must pump more money into the system, even though doing so will only increase the inequality and thus instability! Asset inflation has become an addiction that can only end in disaster, but that you can't stop either.
With scenario (2) in place, the central banks will be able to, basically, bring about a reset of the entire system. By owning a lot of Bitcoin and 'managing' the exchange rate, they will be back in a more powerful position where monetary and fiscal policies will have more freedom to rescue the system even if the lucky decide to cut back spending. The asset inflation can stop, and asset markets will be freer. In the long run, there's really no alternative.