May I include results from a collaboration in my PhD thesis? As a Ph.D. student, I am curious about the following: if you collaborate with a researcher(not necessarily your advisor) while being a student, should you write in your dissertation the work of the other collaborator? 

Assume that while being a student, you collaborate with Prof. X, and you write a paper together. In this paper, the work was split 50-50, so that half of the results belong to Prof. X. 

What should you do when writing your dissertation? Most probably, omitting the results of Prof. X will result in a significant gap in the thesis. I am particularly interested in the case of mathematics: if I cite the result, should I give a proof?
I had a similar situation in my thesis, in math. The third chapter was based on a work that was jointly done with another researcher, a professor from another university. (In the end the paper we published was even with two other researchers, as we decided to join our efforts.)


I had to ask the researcher for permission to include our joint results. This is a no-brainer (though of course he was happy to let me do it). When I sent out the thesis to my committee (of which he was part), I also sent him a copy beforehand and asked him if he was okay with what I had written, as they were his results too and he might want to check that I didn't mangle them before releasing them into the wild. My advisor also was on board from the start, but I imagine that you have to ask for permission if this isn't the case.
I had to include acknowledgments for this in my thesis. Basically, I wrote a paragraph at the beginning of the chapter, stating "this chapter is based on joint work with [...]". Again, this is a no-brainer. I cannot claim all the results are due to me if they aren't. However, I didn't say anything of the sort "I did X% of the work and the other researcher did (100-X)% of the work", this isn't how it works, and it would have been in rather poor taste. At least in mathematics, it's expected that co-authors contribute roughly the same amount of work to a joint paper, but it's not an exact "50-50" (or 33-33-33, or 25-25-25-25...) distribution.
The whole thesis had to be written by me, in my own words, including the results and the proofs that were actually due to the other professor. Moreover, I obviously had to perfectly understand, and be able to explain, and defend, all the results, as if they were mine.


All of this was explained to me by my advisor. It's possible that regulations in your university say something different. The essential part, I believe, which is only hinted at in the steps above, is to ask your advisor what to do exactly.