Take some infinite subset of natural numbers . Define . For some sets A, diverges, like the primes and for other sets, converges, like the squares . One could say the more "dense" a set A is, the more likely it is to diverge. One could also compare the "density" of two subsets of by looking at the asymptotes of their counting functions, i.e. the primes are "denser" than the square numbers because there are more primes under an arbitrarily large x than there are square numbers. But is there such a thing as a "least dense" divergent series?
Question: Let A be a sequence of natural numbers, such that diverges. Does there exist such an A where no matter how we partition it into two mutually exclusive subsets ( but =null) either or must converge? Find such a set A or prove none exists. Conjecture: None exists. For any divergent series A, you will be able to find two mutually exclusive, divergent subsets and .
I think I understand what you are saying. You are suggesting that given an arbitrary set A, define as per your algorithm and is guaranteed to converge. This is true. But what I am asking is to find a divergent set A that cannot be partitioned into two divergent subsets.
Lemma: If diverges, and partition A, then we have one of three cases:
(i) diverges but converges
(ii) converges but diverges
(iii) Both diverge.
What I am looking for is a set A for which no subsets exist satisfying (iii).
Let A be the naturals , which diverge. Letting be evens, and be odds, it is easily shown that case (iii) follows.
Let P denote the primes . Euler and Goldbach independently showed that diverges. Now let be primes of the form and be primes of the form . Since and are equinumerous (Modular Prime Counting Function -- from Wolfram MathWorld), and they cannot both be convergent (as ), they must both be divergent.
Therefore, can you think of a divergent set that cannot be split into two smaller divergent sets like this? *It seems to me that none would exist because just as in the last example, any divergent set can be partitioned perfectly in half, but can this be shown rigorously? If diverges, does that imply that also diverges, for ???
Okay, this would imply that given any sequence of natural numbers for which diverges, it is possible to construct a subset of A, called B, for which the partial sums as L gets large, . Likewise we can find a divergent subset of B that grows approximately as quickly as A and so on. Iterating, given any arbitrarily large integer k, there exists a subset of the natural numbers, whose asymptotic growth is approximately
Therefore there is no such thing as a "least dense" divergent series. A sequence of natural numbers can always be found for which diverges arbitrarily slowly.
The reason I am interested is that it seems to me that the power set of is well-ordered. Though uncountable, you can always take two elements of , say A and B, and say that the partial sums for when L is greater than some arbitrarily large number are:
(i) (A is less dense)
(ii) (B is less dense)
(iii) (A and B are equinumerous)
The same can be said for converging elements of . Now here is the big question: Picking an element at random, what is the probability that it converges? What portion of sequences of natural numbers are "large sets" ( diverges) vs. "small sets" ( converges)? And can we get any kind of handle of a line that can be drawn between them?
There are natural probability measures on (with product -algebra), namely the distributions that consist in picking each number independently with some probability . This gives random infinite subsets of (that have asymptotic density ). Moreover, every natural number plays the same role (the measure is shift-invariant), so that these measures appear to be analogs of a "uniform measure". Choosing would additionally give a symmetry between the random subset and its complement (they would be distributed alike), but I don't need this assumption in the following.
In other words, let and let be independent random variables distributed according to and . The random subset would be . And the question is:
What is the probability that the sum is finite?
The answer is..., wait for it,... zero. Disappointingly. But not surprisingly if you know Kolmogorov's 0-1 law, which tells that the answer has to be either 0 or 1.
I've come up with various proofs but no fully elementary one. One possibility is to use Paley-Zygmund inequality (this is an easy one, don't get scared by the name!) for partial sums to get that, taking a limit, which, given Kolmogorov's 0-1 law, leads to the conclusion since .
Another possibility (without K's 0-1 law) is to write . We notice that almost-surely because of the law of large numbers (Not quite: in fact there is a subtelty, but nevermind, it can be made to work). Then
and almost-surely , so that the right-hand side sum diverges almost-surely as .
As a conclusion: for almost-every subset of (according to the previous measures), . The second proof (almost) only uses the fact that a random set has a positive asymptotic density.
About this "frontier" thing, here is something you might like (but maybe you know it already): for any , the series
(with successively nested logs) diverges, while for any , for any , the series
(only the last function is raised to the power ) converges. For a given , the higher is, the slower the converging series decays. This gives an intuition (not a theorem) of how fast a decreasing sequence should at least decay for making a convergent series.
I had a feeling such. I pictured flipping a weighted coin that came up heads if a number was in the set, and tails otherwise. Regardless of how unfair the coin was, the set would still diverge. If you got heads 1% of the time, you'd get a set containing approximately one of every 100 natural numbers, for example. This would be equinumerous to the set , which diverges, as . I was just having trouble showing this rigorously. This is bordering topology more than my better understood subject of number theory.
Consider this: For some subset of the naturals , define . I believe this is a well-defined homomorphism from . Defining as the set of converging elements of (with the restriction that elements of C be infinite) , we can actually plot C on [0,1] and express the set visually. I believe C is uncountably infinite (proof?) but if the probability that a random element of is in C is zero, then it is a totally disconnected set of infinite points with no "length" . This results in some sort of fractal structure with Hausdorff dimension , doesn't it?
One simple reason why: take an element of , then every subset of is in as well! And the set of subsets of is in bijection with .Defining as the set of converging elements of (with the restriction that elements of C be infinite) , we can actually plot C on [0,1] and express the set visually. I believe C is uncountably infinite (proof?)
This set should indeed have a complex topological structure. If a number is in , then all translates are in . In particular, there are two important subsets of that can be seen as copies of : the one made of sequences not containing 1 (i.e. of the form ) and the one made of sequences containing 1 (i.e. of the form ). Heuristically, these copies have half the (euclidian) size of and there are two of them. I think (not even 10% sure) this should imply that the Hausdorff dimension is either 0 or 1, again... I can't remember enough of fractal theory to conclude...but if the probability that a random element of is in C is zero, then it is a totally disconnected set of infinite points with no "length" . This results in some sort of fractal structure with Hausdorff dimension , doesn't it?
Yes, I neglected to mention that until later in my post. The set is most certainly not equal to , so this homomorphism really only maps , where F represents finite elements of , or terminating decimals. And yes, as F is countable, it does not pose much of a problem, but I do not know enough about topology to correct this flaw.This also amounts to seeing the series of flipped coins as the expansion of a real number is base 2 (1 for heads, 0 for tails). A little care should be taken on "unproper" expansions like 0.0111111... which equals 0.10000..., but this is only countably many points.
I'll have to look into it. It would be impossible to actually "see" this fractal as prescribed here because converges so quickly, but I believe any function of the form would work equally well, slowing the convergence enough to be able to distinguish two points who only differ at, say, the 100th or 1000th digit. Also, I know that if a set is countable, it has a Hausdorff dimension of zero - is the converse true?I think (not even 10% sure) this should imply that the Hausdorff dimension is either 0 or 1, again...
(No answer yet...)
No, it isn't.I know that if a set is countable, it has a Hausdorff dimension of zero - is the converse true?