Co-authorship for grammatical editing

My personal opinion is no, editing at that level is not sufficient.

One can peruse guidelines for your field. Taking Nature as one fairly broad journal, they state:

Each author is expected to have made substantial contributions to the conception or design of the work; or the acquisition, analysis, or interpretation of data; or the creation of new software used in the work; or have drafted the work or substantively revised it

Editing for grammar does not rise to that level for me. And, I can say that over time I have edited a large number of papers for grammar and usage, and never would have asked for authorship. A separate set of eyes looking solely at word usage, spelling, and sentence clarity, while broadly useful, is not at the heart of the work. (Heck, the journal may suggest edits, but would not ask for authorship.)

Such, shall we say, copy editing is quite different from having deep conversations on how to do something and what it really means. While these sorts of interactions can come up, particularly in the early draft stages, they are not about editing, but about analysis and interpretation, not grammar.

No, grammatical corrections do not constitute authorship level involvement with the paper. Some other things (in math) that usually don't constitute authorship:

• Suggesting an example of some interesting behaviour related to a definition or theorem.
• Suggesting a better way to write a proof.
• Pointing out small mathematical mistakes and how to fix them (e.g. missing hypotheses, a remark that isn't true but doesn't otherwise affect the results).

Now, it's possible that when you bring up some of these things to the author(s) that that might spark some discussion of how to implement them and if you are carrying out long, complex, mathematical (not grammatical) discussions with the author(s) that would probably merit co-authorship.

As a rule of thumb, to be an author means that you are claiming responsibility for both the main ideas (e.g. statement of the main theorems and the ideas in their proofs) and the wording of the paper (e.g. you're responsible if your co-author plagiarizes something). Maybe your co-author came up with one idea but then you hashed it out and added your own ideas to it.

Just editing for grammar and spelling you would not be taking responsibility if there is any plagiarism nor are you taking responsibility for the correctness of the theorems nor making any claim that any of the ideas in the paper are your own.

The acknowledgements section is the best way to acknowledge people who contributed to the paper but are not at the level of authorship. You might find that it is worth reading some acknowledgement sections of other papers to find out what sorts of things people give acknowledgement for.

For example, here's the acknowledgement section from one of my advisor's recent papers (https://arxiv.org/abs/1811.04966v2)

We thank Philipp Jell for pointing out Remark 1.10, and Trevor Gunn, Jaiung Jun, Yoav Len, Sam Payne and Thor Wittich for their comments on a draft version of this manuscript. The first author was supported by NSF Grant DMS-1529573 and a Simons Foundation Fellowship.

Philipp provided an example (Remark 1.10). I pointed out some typos and commented a bit on the exposition. And I can't say what sorts of comments the others provided but I will guess they would be similar to my own (comments and exposition).