Sometimes, we want to talk about the ‘size’ of a subset of in a way that’s a bit different from a measure, or indeed cardinality. The following is an elementary but (to my mind) cute series of questions giving the basic properties of this construction, which should be accessible to undergraduates with a basic knowledge of analysis and probability.

Let’s start by considering three sets: the natural numbers , the even numbers , and the square numbers . We’d like to find some mathematical framework that corresponds to the intuition that ‘exactly half’ of is in , and that is, asymptotically, very sparse.

**Things that don’t work.** As any first year mathematician can tell you, the three sets are in one-to-one correspondence, and all contain exactly as many elements as each other. So this doesn’t work!. We also can’t talk about a uniform distribution on either (why?), and so this wouldn’t give us the framework we want.

**Asymptotic (Upper) Density. **Instead, we use constructions called *asymptotic density, *and* asymptotic upper density. *

**Problem 1.** Let us define, for any subset , the asymptotic upper density , and the lower density . Then we can have such that , and disjoint subsets such that .

So these aren’t necessarily the best we can do – we don’t have the nice properties we might ask for. We can do a bit better when we restrict to ‘good’ subsets of .

**Problem 2. **Set be the class of where the upper and lower densities coincide, and for , set . In other words, is exactly the limit , and is the class of where this limit exists. Show that the three sets above belong to , and that we have and .

So this at least gives us a way of formalising the intuition. The next question is to show that we at least mitigate – if not completely avoid the bad behaviour outlined in problem 1.

**Problem 3.** Show that, if are disjoint, then , and that . On the other hand, show that we can have disjoint, such that is not in $\mathcal{F} &bg=ffffff$, and that even if we assume , then we can have a strict inequality .

We’ve now shown that we do a bit better by restricting to sets with a single asymptotic density, where the upper and lower bounds coincide.

The reason I spent a few minutes on a Sunday morning thinking about these objects was a question about constructing sets with a prescribed asymptotic density: given , can we find such that ? The answer is yes, and it’s quite simple to give a (tedious!) constructive proof. However, since I’m a probabilist, I like the following minimal-effort (but non-constructive) solution, which does all simultaneously.

**Problem 4.** Let be a sequence of independent random variables, distributed uniformly on . For , set . Show that .

The fun part here is making the result simultaneous for all , since we have uncountably many events to deal with. It’s probably very straightforward if you have a background in probability, but might be a good exercise if you’re quite new to such ideas!