## Representing points in a set in positional-notation fashion (a trick by Bourgain): part I

If you are reading this blog, you have probably heard that Jean Bourgain – one of the greatest analysts of the last century – has unfortunately passed away last December. It is fair to say that the progress of analysis will slow down significantly without him. I am not in any position to give a eulogy to this giant, but I thought it would be nice to commemorate him by talking occasionally on this blog about some of his many profound papers and his crazily inventive tricks. That’s something everybody agrees on: Bourgain was able to come up with a variety of insane tricks in a way that no one else is. The man was a problem solver and an overall magician: the first time you see one of his tricks, you don’t believe what’s happening in front of you. And that’s just the tricks part!

In this two-parts post I am going to talk about a certain trick that loosely speaking, involves representing points on an arbitrary set in a fashion similar to how integers are represented, say, in binary basis. I don’t know if this trick came straight out of Bourgain’s magical top hat or if he learned it from somewhere else; I haven’t seen it used elsewhere except for papers that cite Bourgain himself, so I’m inclined to attribute it to him – but please, correct me if I’m wrong.

Today we introduce the context for the trick (a famous lemma by Bourgain for maximal frequency projections on the real line) and present a toy version of the idea in a proof of the Rademacher-Menshov theorem. In the second part we will finally see the trick.

** 1. Ergodic averages along arithmetic sequences **

First, some context. The trick I am going to talk about can be found in one of Bourgain’s major papers, that were among the ones cited in the motivation for his Fields medal prize. I am talking about the paper on a.e. convergence of ergodic averages along arithmetic sequences. The main result of that paper is stated as follows: let be an ergodic system, that is

- is a probability on ;
- satisfies for all -measurable sets (this is the
*invariance*condition); - implies (this is the
*ergodicity*condition).

Then the result is

Theorem:[Bourgain, ’89] Let be an ergodic system and let be a polynomial with integer coefficients. If with > 1, then the averages converge -a.e. as ; moreover, if is weakly mixing^{1}, we have more precisely

for -a.e. .

For comparison, the more classical pointwise ergodic theorem of Birkhoff states the same for the case and (notice this is the largest of the spaces because is finite), in which case the theorem is deduced as a consequence of the boundedness of the Hardy-Littlewood maximal function. The dense class to appeal to is roughly speaking , thanks to the ergodic theorem of Von Neumann, which states converges in norm for . However, the details are non-trivial. Heuristically, these ergodic theorems incarnate a quantitative version of the idea that the orbits fill up the entire space uniformly. I don’t want to enter into details because here I am just providing some context for those interested; there are plenty of introductions to ergodic theory where these results are covered in depth.

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