How to improve technical writing

The only way to improve writing (technical or non-technical) is by writing and submitting your work to the criticism of peers. Thankfully, in mathematics there is a culture of blogs. When you learn something new, write it up on your blog and share it with your friends. This will help you better understand what you wrote AND let you practice writing. At first, you won't get much feedback, but as your audience grows you will naturally learn from the feedback they provide.

Another great tool is math.SE and mathoverflow; on these sites you are guaranteed feedback. Ask and answer technical questions, this will let you practice your writing. As your writing and clarity improves you will also notice an average increase in number of up-votes, etc. This will give you useful positive reinforcement.

I was in similar situation than yours during my PhD: my proofs were correct, but tedious to read, and every time one my co-author rewrote them, they looked so much clearer! I wouldn't say that now my proofs are perfect, but they have definitely improved. I would associate this improvement with the following factors:

• my english has improved with time, and I can now use more variations of the "proof vocabulary".
• I have been working with different authors, so I've been exposed to different proof styles.
• I have been writing more proofs, and as Artem's perfectly said, the more you write, the more you get feedback, the better you get.
• I have read and given feedback to other people proofs, so I've started to noticed what I like/don't like on a proof that I haven't written.

So I know it sounds like a dull advice, but I think that it's some skill you acquire with time and experience. Also, there is a wonderful paper from Leslie Lamport: How to write a proof. This paper is not so much about how to write an elegant proof, but rather how to structure your proof. However, once you have a nice structure, I believe it's much easier to make it more elegant. Also, I try now as much as possible to encode my proofs in a theorem-prover (such as Isabelle/Isar), as it helps me understanding for instance what proof steps I can extract and generalize as lemmas.

I haven't read all of them but may I suggest the following:

• Manual for research paper/thesis writers
• How to Write
• Demystifying Dissertation Writing
• Professors as writers