Episode 5: Ponzi Scheme Cont.

So, remember three weeks ago when I mentioned Madoff?

Well, I figured I should let you know in what kinds of infinite worlds Ponzi Schemes will actually work (finally!).

As it turns out, if our world was an infinite lattice, the Ponzi Scheme would still not work. It doesn't work for all infinite worlds. There are restrictions.

A necessary, but not sufficient, condition for a Ponzi Scheme to be successful is an infinite world with exponential growth.

You can see this for the infinite binary tree: there was exponential growth between the points.

We can get even more specific than that, though. We know exactly when a Ponzi Scheme will work for a specific graph (or infinite world). If a graph has no obstruction, then the graph is a Ponzi Scheme.

What does that mean, exactly?

Well, it means that large chunks of the graph have boundary points which are small. So small, in fact, that if k_n is a boundary point for all n, then |2k_n|/|k_n| converges to 0 as n goes to infinity.

Cool, huh?





To think about until next week:
Does every number have a multiple that starts with the number 2004?