The previous post showed how to connect sets with ergodic theory, namely a measure-preserving system , where is the symbolic space and is the shift map. As the reader can check in ERT5, the measure is an accumulation point of Dirac probability measures along increasing intervals of orbits of the point associated to . For this reason, is supported in the orbit of . Then we could take as the orbit closure
instead of the whole space . The set , in addition of composing the mps , has the natural metric induced by . More precisely, endow with the discrete metric and with the product topology. By Tychonoff Theorem, is a compact topological space, and the distance defined as
generates the topology of (to see this, just note that the cylinders – sets of elements with fixed entries in a finite number of positions – form a basis of topology of and each of them is a ball with respect to the metric ). Also, is homeomorphism. In fact, we leave as exercise to the reader to prove that
It is natural to wonder how general results of topological dynamics can be obtained and applied to this setting. This is what we are going to do in this post. The first section consists of the relations between arithmetic properties of and topological properties of . The second section is deeper and we prove van der Waerden theorem assuming a topological multiple recurrence theorem, which will be proved in ERT7. The main result of this post is, then,
Theorem 1 (van der Waerden) If , then some contains arbitrarily long arithmetic progressions.
1. Combinatorics of vs Topology of
(i) is finite if and only if there exist a finite set and an integer such that is the disjoint union
(ii) is thick if and only if .
(iii) if and only if .
Proof: (i) is finite if and only if is periodic for , that is, if and only if there exists such that . Considering , we obtain the desired conclusion.
(ii) If is thick, there are intervals , , such that . Then
which converges to if . For the opposite implication, the same argument works: if then, for every , there exists such that
that is, . As is arbitrary, is thick.
(iii) If , there exist intervals , , such that
Fix an integer and decompose as the union of intervals of length (except, at most, the last one). That is, write , , and into intervals of lenght and one of lenght ( is possibly empty). If is large, and then some is contained in , so that
Again, as is arbitrary, . Reciprocally, if , there exists, for each , an integer such that
that is, . This proves that .
The situation (i) happens if , where . These sets have low complexity and are highly structured sets formed by infinite arithmetic progressions.
Proposition 3 If is minimal, then is syndetic.
Proof: Take any and consider the clopen cylinder
By minimality, the set of return times of to is syndetic. But
and so , implying that is syndetic.
The converse is false. For example,
is syndetic, but contains the fixed point (this follows from Proposition 2, because ). This means we have to look for more conditions about to characterize minimality of .
Question. What are these conditions?
Given , consider the -limit of , defined as
We say that is recurrent if .
Definition 4 A set is called IP-set if it there exists an increasing sequence such that contains the set
Theorem 5 If is recurrent, then contains the translate of an IP-set.
Proof: Construct inductively an increasing sequence as follows: is any element of and, for every , is a positive integer greater than such that the first entries of and are equal, that is
This last condition means that
We’ll prove that . This is equivalent to , for every . The case is obvious:
Suppose for some . Then
which concludes the proof.
Corollary 6 Choose ramdomly, that is, each is in with probability . Then almost surely contains the translate of an IP-set.
Proof: Consider the probability in induced by the vector . By Poincaré Recurrence Theorem (see ERT1), almost-every is recurrent.
2. Proof of van der Waerden theorem
to topology? Well, instead if considering , we take and, for the partition given in (1), associate the element defined as
Such association follows the same philosophy of the previous section: to , there is a natural partition . By the same reasons described in Section 1, is a compact metric space and the same happens to the orbit closure of with respect to the shift . In this encoding, by definition of the distance ,
This condition is guaranteed by the
Moreover, given any dense subset , we can take .
Consider the transformation . By Theorem 7, we can fix such that, for some , the element satisfies