3-4 months ago I found a big mistake in a theorem's proof of a paper that had just been published (stats/cs community). I immediately notified the authors, who claimed to be aware of the mistake but admitted not knowing how to fix it.
This week I noticed that the authors didn't notify the community of the mistake (no updates on their arxiv paper, nothing on the author's websites, etc), even though the paper got a good visibility and already has quite a few citations. I went to check whether the mistake breaks the theorem, and found out that:
1 - the theorem statement is incorrect
2 - however, the statement holds under reasonable assumptions, but the proof requires a different technique (I have a full proof for that)
How should I proceed in this case? This theorem specifically is the main result of the paper, so I wouldn't be comfortable just sending them the corrected proof+statement and not getting properly recognized (e.g. only an acknowledgement). Should I write a short report on post it on arxiv? Send the proof under the requirement that I be added as a co-author? What is the praxis in this case?
Thanks for all the replies so far. I've contacted my advisor and have finished writing a short report (~4 pages) on the matter. I'm yet to decide whether I should contact the authors before posting the report, and whether a merger would be a good idea -- as some people said, it is a huge red flag that they admitted to be aware of the mistake and never took action, so I am not certain that I would like to be associated with the authors in a collaboration level.
1) The paper has already been published in a conference and its journal proceedings. I've contacted the authors after acceptance but before the proceedings were published, and they took no action.
2) The paper proposes a new algorithm B and states that it has the same running time as algorithm A, a classic method in the field, while having smaller memory cost and being easy to implement. The main theorem roughly states that T(B) = O(T(A)), which is wrong as for some inputs T(B) = infinity. The technical mistake is subtle and involves upper-bounding an infinite series, which in reality can diverge to infinity. I've ran the algorithm on a simple instance where it does not halt nor makes any progress.
3) It can be shown that, for some input distributions, T(B) = O(T(A)). The technical argument is a bit different from what they initially presented. Unfortunately, I think that for these distributions it can be shown that A has an even smaller memory cost than B, but I am not sure.