I hope that this question is on-topic, though it is not quite technical.
I am curious to hear from people how they approach reading a mathematical paper.
I am not asking specific questions on purpose, though at first I had a few. But I want to keep it rather open-ended.
I have a couple of pieces of advice:
Do not be intimidated by the introduction. This is where the whole paper is laid out,so it typically won’t be easy to follow. The author will expand sentences into whole pages later on.
Take notes. I usually get a legal pad out and start copying down the paper, and fill in things, and make notes to myself or work out small examples.
Talk to people about the material. If you have an adviser or someone you are going through the paper with it is often helpful to present portions of the paper if not all of it.
Skip confusing bits. Feel free to black box certain parts. If you get stuck on something, don’t worry, one sentence should not keep you from progressing through the paper. Box it off, pretend you understand the conclusion and make a note of it. Take that piece as a fact and move on to the next part of the paper. The goal is to get something out of a paper not understand it line for line the first time through.
Come back to the paper. You may want to go over the paper multiple times, take several passes at it. At least this is what I have to do when I am reading something. The amount of time it takes me to assimilate different levels/types of arguments gradually decreases, but it does take time.
I don’t spend a lot of time reading papers right now, but this is what I try to do if I really want to understand something.
Edit: It has become easier and easier for me to read and digest papers over time. I am still slow, but I can get through a lot more. The above process is sort of a set out routine to help you digest it, eventually it will become obsolete.