I want to implement, in my app, a technique that has been patented in the USA. I live in Bulgaria. Do I need permission from the patent owner?
Bulgaria is one of many countries in which intellectual property can be protected with a single international patent filing, in lieu of a Bulgarian specific filing. In generally, you should assume that any patented idea likely to have international application, such as Internet based software, which is patented in the U.S. is also the subject of international patent protection, unless you specifically know that this is not the case. Virtually all patent lawyers know how to make this filing and most would recommend doing so in this case.
So, yes, it is something you should worry about, and the better practice, by far, would be to obtain permission from the patent owner if the app is likely to have any meaningful commercial value.
Now, if you were sued, there is a good chance that it would be in Bulgarian courts under Bulgarian patent law, which may be somewhat less harsh in some respects like damages awards than U.S. patent law. But, the basis for patent liability for an infringement in Bulgaria would be very similar.