"there are fewer than 1,000 billionaires and the vast majority of them are self-made"
That's a refutation, Piketty made assertions not just about the outcome, but about the process by which that outcome would happen.
It doesn't change the thesis, which is wrong. r is not greater than g for any reasonable amount of time for individuals/families because humans have an amazing ability to behave stupidly and reduce r below g.
But inherited wealth does not dominate wealth amassed from a lifetime's labor. The largest fortunes today are dominated by people who built businesses, not by people who inherited their wealth. So if "there are fewer than 1,000 billionaires and the vast majority of them are self-made", that is a direct refutation of Pikkety's thesis.
It is hard to overstate just how bad Pikkety's arguments are. His argument assumes single inheritance of fortunes, it assumes that wealthy people don't donate significant sums to charity, it assumes that inheritors of wealth will invest that wealth as wisely as the person who created it, along with many, many other false assumptions. The most egregious one in my opinion is that it assumes the wealthy don't spend their money.
If a billionaire makes a 5% return on his billion dollars and spends $50 million dollars a year, does wealth inequality increase? No, it does not.
It's a ridiculous theory, it is directly contradicted by the facts, and I have no idea why it is taken seriously by people here other than just as an opportunity to participate in a 2 minutes hate against rich people.