I wrote to math-fun back in January about a way of measuring sets that (like cardinality) makes {1} bigger than the empty set, and (like density) makes {1,2,3,4,...} bigger than {2,4,6,8,...}. (If you know of other theories with this pair of properties, please let me know!) Up until recently the only haven for this theory would have been the nonexistent journal "Definitiones Mathematicae". But now I have actually proved something nontrivial about my definition, or at least provided a non-trivial proof of something. (It's possible that my result is trivial and I just don't know it!) Anyone who's interested can check out "One-Dimensional Packings: Maximality Implies Rationality" ( https://arxiv.org/abs/1704.08785), submitted to Discrete Analysis. Here the word "maximal" carries a hidden freight, namely, my notion of what it means for one set of natural numbers to be "bigger" than another. I would've been happier with an article that actually proved that maximal packings (always) exist, and that they're unique, and that they're rational (i.e., ultimately periodic), as opposed to merely proving that *when* they exist, they must be rational. But I'm thinking that the best way to get other people to take an interest in these matters is to start publishing *something*, even if it's on the weak side. Can any of you can see a way to prove that maximal packings always exist, and that they're always unique? Also please let me know if you're aware of any literature on one-dimensional packings (perhaps by you!) that I should cite. Jim Propp On Mon, Jan 30, 2017 at 12:30 PM, James Propp <jamespropp@gmail.com> wrote:
If S is an eventually-periodic subset of {0,1,2,...}, then |S|_q, defined as the sum of q^n as n ranges over S, is a rational function. The map from S to |S|_q is a finitely-additive measure ("valuation" is probably a better word to use) taking its values in the ring of rational functions in q. If we order that ring "at 1", we get a way to compare sizes of sets of natural numbers. In fact, it is a total ordering (though not a well-ordering) of the set of eventually-periodic subsets of {0,1,2,...}.
This may remind some of you of the ordered fields that you get by adjoining to R a formal infinitesimal or a formal infinity. In both cases, you've got the field R(x), and all that differs is the ordering you put on it. My proposal is similar, except that I'm ordering the field at 1 rather than 0 or infinity.
Writing |S|_q as A / (1-q) + B + lower order terms, we can focus on just A and B. A measures the density of S, and B captures finer information. The first thing to notice about B is that it's sensitive to perturbations that don't affect the density; if you add an element to S, you increase B by 1, and if you remove an element from S, you decrease B by 1 (even though A doesn't change).
But B conveys subtler information too:
For instance, compare {0,2,4,...} with {1,3,5,...}; the former has B = 1/4 while the latter has B = -1/4. Informally, we say that sliding the set of even nonnegative integers over the right by 1 reduces the "size" of the set by 1/2. And that makes sense, because if you slide it over to the right again, "losing" half an element again, you get {2,4,...}, which is missing precisely 1/2 + 1/2 = 1 of the elements of {0,2,4,...}.
(Yeah, this way of measuring sets is not translation-invariant. But you gain something nice in return: a proper subset of a set is always strictly smaller than the set.)
I recently learned that Ilya Chernykh has been exploring the same circle of ideas (see http://mathoverflow.net/questions/215762/non-standard- numbers-and-exponential-form-of-zeta-function), but I don't know of anyone else who's fished in these waters. (You could argue that analytic number theorists know all this already and don't think it's interesting.)
Anyway, the map that sends S to (A,B) should I believe be called a "Grandi valuation", since it gives a rigorous framework for making sense of Grandi's formula 1 - 1 + 1 - 1 + ... = 1/2. (Indeed, Bernoulli's formula 1 + 0 - 1 + 1 + 0 - 1 + ... = 2/3 can be understood in the same way: {0,3,6,...} exceeds {2,5,8,...} by 2/3 of an element, if we measure size using this valuation.)
More precisely one might call the map the "Grandi-Abel valuation", since it uses Abel regularization of the summation of the indicator function of S. Anyway, other ways of taming divergent series exist, so there will be other Grandi valuations. And maybe the term should also be applied to the non-truncated valuation (the one that takes values in the ordered field R(q)), which gives the sets {0,3,4,7,8,11,...} and {1,2,5,6,9,10,..} different "sizes" (the former is infinitesimally larger). In the non-truncated valuation, we get a translation-covariance property that might console us for the loss of translation-invariance: shifting a set corresponds to multiplication by q.
One can try to extend the Grandi valuation to sets that aren't eventually periodic, as I do in my MathOverflow post, and to some extent this works. However, one can construct sets that are incomparable, so if you're a fan of number systems that satisfy trichotomy, it's important not to overreach.
I haven't given a definition of what a Grandi valuation is in general, but a hallmark is that a Grandi valuation should assign to any eventually periodic set S a value in some module from which the quantity I called "B" can be derived.
Anyway, I have a bit of a crush on this notion, and I'd like to put it to some sort of use. My current target is a general uniqueness theorem in the context of one-dimensional packing.
Jim
P.S. It has occurred to me that when we measure infinite sets in this way, a cute name for this kind of generalized cardinality would be "grandiosity". But I think it's prudent to avoid introducing this term until I have a better idea for how it should be defined!