A ⊂ ℕ is such that Σ_{n ∈ A} 1/n diverges
is one way of saying that the set A of integers is not too sparse.
The set of powers of 2 does not satisfy this condition ... it's too sparse.
The set of primes does satisfy this condition ... it's not too sparse, primes turn up "reasonably often".
We are looking for set of three numbers that are "equally spaced". So {4, 7, 10} are equally spaced, differing by 3 each time. Another set might be {20, 30, 40}, this time differing by 10. We'll call such a set "Equally Spaced Triples", or "EST" for short.
If you have the positive even integers - 2, 4, 6, 8, ... - then clearly you can find infinitely many ESTs. You have {2,4,6}, {6,10,14}, and so on. However, we can show that if you take the powers of 2 - 1, 2, 4, 8, 16, 32, 64, ... - then we cannot find an EST.
So, when can we do this? When can we be guaranteed always to find infinitely many ESTs? Suppose you have a set of numbers - n0, n1, n2, n3, n4, ... - is there a test to see if we are guaranteed to have infinitely many ESTs?
The answer is yes, and that's the result that has been proved. The result says this:
Take any set of positive integers, take their inverses, and add them all together. If the result has an upper bound, then the set might not have infintely many ESTs. However, if the sum grows without bound, then you are guaranteed to have infinitely many ESTs.
Unpacking that with our examples, taking the inverses of the powers of two and adding them up we get 1/1 + 1/2 + 1/4 + 1/8 + 1/16 + ... and we can show that the total never exceeds 2. In fact, the total never reaches 2. So since the total is bounded, we do not have infinitely many ESTs.
Now look at the primes. There is a standard result that says that the sum 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + ... is unbounded above. You give me a desired total, and I can tell you how many terms you need to take to exceed that number. So the sum of the inverses is unbounded, and hence the primes will have infinitely many ESTs.
So, in summary, if a set of positive integers is dense enough - if there are enough of them in some technical sense - then there are infinitely many ESTs. The test for density is to ask that the sum of the reciprocals (inverses) is unbounded.
Does that help?
Edit: I wanted to contact you out-of-band, but you only have a LinkedIn link in your profile, and I don't use LinkedIn. If you're interested in discussing this further then I'd be happy to help, but better by email. My contact details are in my profile.
Edit 2: Thank you everyone for your kind comments. You've made me think about my write-ups. I already do a lot of writing ... I might re-visit what and how. I appreciate the kind words.
I would like you to know that I would gladly pay a monthly fee to have at least 1 proof (per month) explained to me like this. Not sure how that could scale to varying knowledge levels, but I would love to have a slow educational drip of math explanations.
Even a podcast would be awesome.
Have you subscribed to sixty symbols on YouTube or other math related channels? I find that helpful.
Not math-focused, but you might love The Morning Paper:
Yes, as you say, the "unbounded sum" condition is sufficient, but not necessary. An example that's obvious "by inspection" is just to take an EST for every value of 2^n ... so take (2^n), (2^n)+1, and (2^n)+2.
Also updated my profile
Of course, a perfectly reasonable stance is that if someone really wants to contact you then they'll do their homework and solve the riddle, in which case it's fine. I just thought I'd let you know that not everyone lives in the world where that kind of knowledge is commonplace.
Updated. I hope that’s better. I had just saw your profile and saw all the spam you got and thought I had to come up With a really obscure way to obscure it. Sorry.
Would you care to clarify? If it's a question then I'll try to answer, but if it's a conjecture, can you make it more precise?
I don't have an answer, it's not my area, but it does feel a lot like the condition "sum(1/n) diverges" is likely to be close, but that's just a gut feeling.
It's probably also worth noting that weird density distributions are possible. E.g. one could imagine a kind of oscillation where you include enough values to get the total density up to that point up to some decreasing threshold (e.g. 1-2^-k if we've flip-flopped k times) then omit values to get below a different threshold (e.g. 2^-k if we've flip-flopped k times), and repeat the process indefinitely. The count up to a point n for this construction has the interesting property that it isn't greater in a big-O sense than the count for exponentially sparse sets while also not being less than the count for any sublinear density function in a big-omega sense (using the Knuth interpretation).
If you have a sequence that grows asymptotically like n^m it means that f(n)/n^m goes to a constant asymptotically. So the question is if this implies that sum 1/f(n) converges. I feel intuitively that it should be possible to prove that.
A sketch: For every epsilon there is an N such that |1/f(n) - 1/C n^-m| < epsilon n^-m for n > N. So if we take the difference between the reciprocal partial sums, then that is bounded by eps * the partial sum. That is finite exactly if the reciprocal sums converge. Thus the convergence behaviour is the same for this case...