One of the first astonishments I faced as an undergraduate mathematics student was that chaos prevails. What do I mean by that? Well, there are more irrational numbers than rational numbers, there are more non-computable functions than computable, and, finally, there are more continuous functions devoid of a derivative than those which have one. In the light of a myriad of examples alike, one might be easily persuaded that most of the mathematical objects are ill-behaved.
In my final undergraduate years, I came to realize this is not the case. In particular, using robust measurements, one can show that the set of somewhere differentiable functions in is so thick that any compact set can be shifted in such a way that it becomes a subset of somewhere differentiable functions. This is one of the results obtained in my joint work with Adam Kwela: Differentiability of continuous functions in terms of Haar-smallness.
Thickness implies that the underlying set is not Haar-null nor Haar-meager. Yes, it is what it sounds like, a generalization of a null set and a meager set. If we restrict ourselves to the subsets of the real-line, these notions are equivalent.
- A null set is a set with a zero length, area, or volume.
- A meager set is a set that is a sum of a sequence of sets of which closure has an empty interior.
- An example of a meager set that is not null is a fat Cantor set.
- An example of a null set that is not meager is the complement of the sum of Cantor sets with measures and so on.
Consequences of the thickness of the set of continuous multi-dimensional somewhere differentiable functions are particularly interesting. In our work, we proved that Banach-Mazur theorem generalizes to higher dimensions. Therefore, two initially similar (or even equivalent) notions diverge as we pass to infinite dimensions, i.e., to the set of functions. The set of continuous multi-dimensional somewhere differentiable functions is, on the one hand, meager, and yet, on the other, it is not Haar-meager.
Another corollary that raises an eyebrow is a sharp contrast with the one-dimensional case. The space where only one lonely plays a role is small in every reasonable sense. Here, the set of nowheredifferentiables is meager, Haar-meager, Haar-null, and what not! Adding a little -buddy changes everything and we are left with meagerness only.
Now, I want you to take this with a pinch of salt. Chaos is prevalent in the world of one-dimensional functions. However, what is ill-behaved in the 1D world is not so thrilling once we look at it from a 2D or 3D perspective. Considering functions in terms of more than one variable seems to be a more “true” way of looking at these objects. After all, the physical universe is described by at least four dimensions: height, width, depth, and time (and String Theory goes way beyond that!). So, here is the food for thought, maybe chaos is not so ubiquitous in the mathematical realm?