I am currently finishing a paper, and I am writing the acknowledgements section.
There is a colleague with whom I have discussed several times about the question answered in the paper, who certainly spent some spare time thinking about it, and suggested some interesting ideas. However, none of these ideas actually turned out to be fruitful. While I appreciated their interest in this question, and things definitely might have turned out differently, the paper as it is has not actually benefited in any tangible way from my discussions with this person.
Do you think it is appropriate to thank them anyway? (And, if yes, with what formulation? just the standard "I would like to thank Prof. Smith for some interesting discussions", or something else?)
If it matters, the field is pure mathematics.