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I'm writing my first paper and my last (main) result has a very nice intuitive reasoning, but my rigorous proof has turned into an ugly 4 page epsilon argument. The paper is on a topic in number theory (in particular showing the upper density of a certain set of integers). I have currently chosen the following plan of attack:

For every $\epsilon > 0$ I show there exists $N$ large enough s.t. 6 different quantities are all $< \frac{\epsilon}{6}$. Currently I have the bounds on the 6 different quantities as a 6 part lemma, where I find $N_1,N_2,...N_6$ s.t. for $n > N_i$ the i'th quantity is $< \frac{\epsilon}{6}$. I'm not concerned with how large my $N$ is, just that there is a sufficiently large one.

I'm quite unhappy with how my argument reads. The insight as to whats going on is completely lost in the equations. I tried compensating by giving a short paragraph explaining the reasoning behind the equations, but the reader will still have to toil through 4 pages of equations.

I was hoping to see some links to well written papers that have to deal with a similar situation that I'm in, preferably in the fields of number theory or combinatorics. I have entertained the idea of switching to big-O notation for everything and avoid the epsilon argument all together, but not sure how I'd be able to make that work. Any advice would be greatly appreciated MO!

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    $\begingroup$ This might not be at all relevant to your paper, but Terence Tao had a nice (actually, more than one) article on epsilon management: terrytao.wordpress.com/2007/06/25/… $\endgroup$
    – Todd Trimble
    Commented Sep 29, 2011 at 13:29
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    $\begingroup$ Just put the intuitive paragraph first and then give the details. The reader can choose to skip the horrible computations if she wants. $\endgroup$
    – Andy Putman
    Commented Sep 29, 2011 at 14:35

4 Answers 4

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Proposition 1: For sufficiently large N, the following six quantities can be made arbitrarily small:
a) b) c) d) e) f)

Proof: The proof, which is something of a technical distraction, is deferred to the end of this section.

Theorem: [Insert great theorem here.]

Proof: [Clear intuitive proof, invoking Proposition 1.]

It remains to prove Proposition 1.

Proof of Proposition 1: [Insert long boring computations here.]

Or alternatively:

Theorem: [State theorem]

Proof: I claim that for sufficiently large N, the following six quantities can be made arbitrarily small: a),b),c),d),e),f).

Granting this claim, the proof proceeds as follows: [intuitive argument here].

It remains to prove the claim:

Proof of claim a):

Proof of claim b):

Etc.

Edited to add: Also: There is absolutely no need ever to write the expression $\epsilon/6$; that's for students who are proving to their instructors that they understand what's going on. In a research paper, if you prove that six quantities can all be made arbitrarily small, you can safely assert that their sum can be made arbitrarily small and count on your readers to understand why.

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    $\begingroup$ +1, although I prefer the first outline to the second. $\endgroup$
    – Theo Johnson-Freyd
    Commented Sep 29, 2011 at 6:35
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    $\begingroup$ Indeed, I have read (and, inspired by them, written) papers in which all proofs are deferred to the end. It goes: Section 1: Introduction. Section 2: Statements of all definitions and results, explaining the flow of the overall argument but none of the details. Sections the rest: Proofs. $\endgroup$
    – Theo Johnson-Freyd
    Commented Sep 29, 2011 at 6:36
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    $\begingroup$ I recommend giving the proof in some correct form. This is not an announcement but a published account of the result, which should therefore be proved. A proof is not as boring to someone who wants to know why it is true as you might think. Write it up as well as you can and then let the referee tell you how to improve it. $\endgroup$
    – roy smith
    Commented Sep 29, 2011 at 6:52
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    $\begingroup$ I like you answer: splitting things into small pieces which can be understood easily is always a good strategy. But in any case, one should really give enough information/proofs that the whole thing is userfriendly. Nothing is more annoying that a paper where you have to do all by yourself ;) $\endgroup$
    – Stefan Waldmann
    Commented Sep 29, 2011 at 7:40
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    $\begingroup$ I hate the system (especially popular in computer science) where all nontrivial proofs are deferred to an appendix, similar to what Theo describes. It makes it quite hard to read, as one has to constantly switch between the appendix and the main body of the paper, it makes the text fragmented and it is annoyingly distracting. Proofs should be given right after the theorems they prove, in as much linear fashion as possible. Swapping a technical claim with the main part of a proof of a major theorem as Steven describes is OK, but moving proofs further away is evil. $\endgroup$
    – Emil Jeřábek
    Commented Sep 29, 2011 at 9:17
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(a) Don't forget that there is a notation $o(1)$ for functions whose absolute values are eventually less than any positive $\varepsilon$.

(b) These days there is an option for proofs that are technically long and tedious. You can put the full proof on the arXiv, and publish an expository outline of the proof in a journal. The journal article can cite the arXiv article. This can also help you to get sympathetic consideration from the journal and its referees.

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    $\begingroup$ I have doubts whether this solution promotes a good writing and publishing culture. Chances are few people will bother to read the "full version" and therefore one loses an incentive to polish the proof. Splitting paper into two seems a bit artificial to me. In programming, when you have to structure your code in artificial way, there might be something wrong deeper in the flow of your program. If you cannot structure your paper in a way that is readable, maybe it means you should rethink the proofs/try to improve the flow of the argument. $\endgroup$
    – Marcin Kotowski
    Commented Sep 29, 2011 at 18:27
  • $\begingroup$ I don't see that having details in an arXiv paper that few people read is worse than having a long section in a journal that few people read. Some proofs are long and tedious no matter how hard we try to improve them. A real-life example of where a journal liked this arrangement was where a proof had some 20+ pages of technical analysis that were more or less along the same lines as previously published proofs. The journal wasn't happy about publishing 20 pages of calculation with no innovative ideas in it, and yet the theorem that relied on it was great. The compromise made everyone happy. $\endgroup$
    – Brendan McKay
    Commented Sep 30, 2011 at 0:40
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Sometimes fiddly $\epsilon$ bounds can be eliminated by carrying out the proof using nonstandard analysis. (You probably don't want to learn about nonstandard analysis just to rewrite one proof, but if you find yourself in this situation repeatedly, it might become a worthwhile investment.)

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  • $\begingroup$ Presumably, you then want to quote a general result that your proof didn't really depend on the extra weird axioms of non-strandard analysis (such as the existence of a real number that is smaller than 1/2, smaller than 1/3, smaller than 1/4, smaller than 1/5, smaller than 1/6, ...) Is there such a metatheorem? $\endgroup$
    – André Henriques
    Commented Sep 29, 2011 at 15:59
  • $\begingroup$ @André: Certainly, there are such metatheorems. Details depend on what kind of setup one uses for nonstandard analysis, but generally speaking, nonstandard analysis is a conservative extension of standard mathematics (meaning that you cannot prove any new theorems that only mention standard concepts). $\endgroup$
    – Emil Jeřábek
    Commented Sep 29, 2011 at 16:10
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    $\begingroup$ I would avoid the use of mathematical theories like logic just for the sake of superficially shortening an exposition. While I assume that none of the theorems in logic required to prove that NSA is conservative over ZFC is actually difficult to prove, logic itself is not at all easy to learn. (What makes a field of mathematics difficult to learn aren't the hard proofs, but rather the notations, the abuses of notation, and the unwritten intuitions; all of these are rather complex in mathematical logic.) Use of mathematical logic seriously narrows the audience of your paper. $\endgroup$
    – darij grinberg
    Commented Sep 29, 2011 at 19:21
  • $\begingroup$ (I have a similar problem with people using sheaves and schemes for no reason...) $\endgroup$
    – darij grinberg
    Commented Sep 29, 2011 at 19:21
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I believe long tedious boring calculations should not be put into papers. I would try to describe exactly how to do the verification, but leave the details to the reader. If there are steps that need a non-trivial idea, I would indicate the idea needed, but avoid the standard calculations.

My rule of thumb: write down exactly enough so that you yourself could reconstruct the argument in a few years, but nothing more.

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    $\begingroup$ A few years?! Is that a typo? $\endgroup$
    – Mark Grant
    Commented Sep 29, 2011 at 6:52
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    $\begingroup$ Write down enough so that if you looked at the write up again, a few years later, it would be possible to check the details yourself again without getting stuck. $\endgroup$
    – Alexander Moll
    Commented Sep 29, 2011 at 6:57
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    $\begingroup$ I agree with your rule of thumb. Nothing is more annoying than going back to an old paper of oneself and find that this stupid old bast*** write in such an arrogant way that one does not understand ones own reasoning frmo 5 years ago anymore. I guess, if I have problems with my old papers, then others will have them, too, and that is very bad. Of course, there are standard kind of computations/estimates etc which everybody gets done himself, but still, one should try to be "userfriendly". IN addition to your rule I would add: bring the details but in an organized way. Try to split things up... $\endgroup$
    – Stefan Waldmann
    Commented Sep 29, 2011 at 7:37
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    $\begingroup$ I will make sure not to read any of your papers. Avoiding "standard computations" is how bugs are introduced. $\endgroup$
    – Igor Rivin
    Commented Sep 29, 2011 at 9:41
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    $\begingroup$ I just hope I don't get such a paper to be a referee! $\endgroup$
    – Reimundo Heluani
    Commented Sep 29, 2011 at 12:01

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