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I have sent one of the Ph.D. (2019) chapters to a journal two months ago. Today, I found a preprint (Feb 2021) of one of the strongest key authors of my field. Her preprint talking about a similar topic to mine, which I have done two years ago. I send it to multiples journals. Each journal wastes my time for about 3 to 4 months. She is a famous author and may her paper will be published before mine. What should I do now? Do I need to send a letter to the journal editor telling them about this good point about my project. If so, what should I told them.

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    I advise you to not send any email.
    – Alice
    Mar 7, 2021 at 10:30
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    It's perfectly fine for two papers to "talk about the same topic". Knowledge would not progress otherwise. Mar 7, 2021 at 10:31
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    Yes, that will change nothing. Just be patient.
    – Alice
    Mar 7, 2021 at 10:35
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    Perhaps upload a preprint of your own to establish priority?
    – Allure
    Mar 7, 2021 at 12:23
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    In the ms, if not done yet, refer to your thesis.
    – Alchimista
    Mar 7, 2021 at 15:29

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This question was asked during the pandemic and never received an answer, but did receive some helpful comments. I think the question represents a common enough situation. I will try to provide an answer, to get this question off the unanswered queue.

Probably the answer to this is very field dependent. In mathematics, many people upload preprints to arXiv as soon as the paper is ready to be submitted, which establishes priority. In the case of a senior person X overlapping with a PhD thesis, usually the advisor can write to X to explain the situation, and X will try to make sure your work can still get published. No one wants to kill a PhD student's career.

In your case, since X's work is already available as a preprint, it would be appropriate to send them a copy of your work (if your advisor agrees of course), discuss potential overlap, etc. This also has the upshot that X can update their paper to cite yours and explicitly say that yours was first. Again, the advisor can help with this email exchange. You're not in a position to make demands, but usually the bigshot will suggest citing/advertising your work once they realize the level of overlap.

Hopefully your paper gets accepted by the journal where it's currently under review. If not, the editor of this journal might be able to smooth the process at another journal, also explaining the history to the next journal so that they don't hold it against you that X has now done some of the same work. It's pretty common for two papers to do something similar at the same time, and both get published. Usually there are some differences in approach, and in the long run both will get cited. Even if X's version does get published before yours, people will still know that the publication timeline is wonky, especially if you update your paper with a line about how this work was done in 2019 (and you're grateful to your advisor, etc.), and you subsequently learned about X's work in 2021, and you refer the interested reader to that paper for a treatment of a similar problem, etc.

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