How much comparing to another paper is too much?

Question

As a new researcher, I am in the following situation in mathematics research:

I read paper X, a short paper published in a low-mid tier journal, and found a way to improve and extend the result. The technique I used to extend the result is a different approach to the problem, but not that mathematically technical. However, even though the mathematical extensions are (arguably) trivial, the extensions open the door to a much broader approach to my field. They also open the door up for interesting simple examples that were previously unable to be produced.

I am currently writing up my findings of the original extension in a paper. In this paper, I correctly reference paper X when necessary. From an ethical point of view, it is crystal clear what I did and didn't do.

1) Is it bad to compare to one single paper often in a paper? 2) What is a good way to tell whether a result is incremental or not? 3) Some of the arguments in paper X need slight modifications under my extension. Is it okay to repeat some of their ideas in my proofs (with clear citation/credit of course)?

Another problem arising from 2): I have another, much more interesting result because of this extension that I have not published. However, this result moves in a different mathematical direction and therefore, I sort of want to write a separate paper on it. I am faced with the following dilemma. I could reasonably do one of the following:

I) Write a longer paper and work to bring the ideas together. Pros: My separation from paper X is much more obvious and seen as less incremental. Cons: I mix two different ideas and risk the paper telling too many stories.

II) Vaguely hint at the second result in the first paper. If I do this, the problem is that if my hint is too vague, it will make my first result seem uninteresting. If it is too specific, I risk showing someone else my idea and having them "beat me to the punch." Pros: Sticks to one story, but provides additional motivation. Cons: May make the first result seem too weak.

I apologize if this post is nonspecific, but I imagine others have faced this problem in their relative fields. How did you resolve this dilemma? I realize I have to figure out the answers myself, but how did you figure out the answers to these questions when you were facing this dilemma yourself?

Thank you.

Answer 1

1) Is it bad to compare to one single paper often in a paper?

Not at all. It happens all the time. If it represents a significant improvement of the paper in question, especially one that opens up new possibilities, you don't have to hide the fact that this is what you have done. The same answer applies to your question 3). Whether or not it is a significant improvement of the paper (and thus the answer to your question 2) is exactly the sort of thing that peer-review is supposed to address. Trust your instincts. Resist the temptation to inflate what you have just to get a larger paper.

As far as the second set of questions go, you could write two papers and submit them simultaneously, the second referencing the first. It isn't uncommon for researchers to cite unpublished but submitted work.