Did you know that US has tools to claim part of your wealth even if you relinquishment the citizenship. Google "exit tax" for us citizens.
The US exit tax is a tax on your worldwide assets. The tax applies to all property that you own on the date of renunciation
Most of the times I've heard of it happening it's someone who's net worth is well below the threshold for this (indeed well below the threshold where they actually need to pay any income tax to the US government), but because of the headache of needing to file the paperwork and the fact that a lot of banks don't want to deal with US citizens due to extra requirements imposed on them by the US government (via its financial system).