I have the following question:
How likely it is that an author carefully read through a paper cited by him?
Not everyone reads through everything that they have cited. Sometimes, if one wants to use a theorem that is not in a standard textbook, one typically finds another paper which cites the desired result and copies that citation thereby passing the responsibility of ensuring correctness to someone else. This saves a lot of time, but seems to propagate inaccurate citations and poor understanding of the work being cited.
The question is thus about what should authors' citing policy be, and to what extent authors should verify results they are citing rather than using them as black boxes.
Best Answer
I normally do. Right now I'm facing a tough choice though: to read David-Semmes book in honest or to write something like "We prove that A implies B. The reader can juxtapose that with the claim on page ... in [] that B implies C" instead of "We prove that A implies C" in the introduction. Being as lazy as I am, I am inclined to go for the second option but that will certainly make the paper less "sexy", so my co-authors do not feel very happy about it.
However this shows how you can avoid both reading the papers you refer to and the uncertainty about whether what you declare proved is actually proved: separate the part you prove from the part you refer to in a crystal clear way and take credit for the reduction only rather than for the full statement (which, frankly speaking, is as much credit as you can really claim anyway).
The correction mechanism Nik mentioned works primarily in the way that most things just go unnoticed because nobody reads those papers or uses them in any way. When something is really important, it gets a lot of attention and somebody finally straightens things out. However, that doesn't happen fast and I have learned it hard way. My 2002 Duke paper joint with Treil and Volberg on the system Tb theorem has an error in the proof. It had been cited a lot of times before the error was finally spotted and corrected by Tuomas Hytonen around 2010. This also shows that an erratic argument isn't always useless or fatally flawed. Sometimes it is just an "incomplete proof". To my shame, I should also mention that it was one of the cases when I didn't read the final draft carefully and relied on my co-authors to do that. Apparently, they had a similar attitude...