In this post I explore the notion of piecewise syndeticity and its relation to topological dynamical systems and the Stone-Čech compactification. I restrict attention to the additive semigroup but most results presented are true in much bigger generality (and I tried to present the proofs in a way that does’t depend crucially on the fact that the semigroup is ). These results and observations are collected from several sources, and are scattered in the literature so I thought it would be nice to collect them all in one place. Probably the union of Furstenberg’s book with the book by Hindman and Strauss contains most (if not all) of the things in here; these surveys of Bergelson are also a good source.
— 1. Introduction —
There are several notions of largeness for subsets of , each notion with its advantages and disadvantages. One such notion is that of syndetic set, i.e., a set with bounded gaps. More precisely, is syndetic if there exists a finite set such that any natural number can be written as with and .
Example 1 The following are syndetic sets:
- The set of all natural numbers,
- Any co-finite subset ,
- The set of even numbers,
- Any infinite progression of the form ,
- The set for any , where is the distance between the positive real number and the lattice of natural numbers.
- The set for any and any with an irrational coefficient (other than the constant term).
One problem with syndeticity is that this notion is not partition regular, i.e. if is a syndetic set and we decompose , it may happen that neither nor are syndetic (take and and consisting of increasingly long alternating sequences of intervals). One way to fix this is to consider its dual family and then its intersection family (spoiler alert: we end up with piecewise syndetic sets).
Definition 1 A set is called thick if it contains arbitrarily long intervals. More precisely, if for any finite set there exists such that .
It should be clear that it is possible for a thick set to be decomposed into two sets with neither nor being thick.
Example 2 The following are thick sets:
- The set of all natural numbers,
- Any co-finite subset ,
- The set where and is the Fourier transform of a non-atomic measure on .
- The set for any with .
It is not hard to check that is thick if and only if for every syndetic set . Given a family and its dual family it is natural to consider the family of intersections:
Definition 2 A set is called piecewise syndetic if there exist a syndetic set and a thick set such that .
Example 3 Any syndetic set is piecewise syndetic and any thick set is piecewise syndetic. My earlier post listed several sets which are NOT syndetic; it turns out that the same proofs show that those sets are not piecewise syndetic. These examples (of sets which are not piecewise syndetic) include:
- The set of prime numbers.
- The set of square-free numbers (and more generally, given , the set of numbers not divisible by any power with prime).
- The set of numbers with at most distinct prime divisors, for a fixed but arbitrary .
- Let . The set of numbers such that when is decomposed into prime powers, there is no exponent equal to .
It turns out that a family of intersections must be partition regular:
Whenever a set is decomposed into finitely many pieces , some piece is also in .
Proof: It suffices to show the result for ; the general case follows by induction. Assume that , where and is decomposed with . Let . It is easy to check that , so if then and the proof is finished. But if, on the contrary, , we have . Finally observe that , which shows that in this case .
In particular we obtain the following classical result.
The following description of piecewise syndetic sets is not surprising but sometimes useful.
Let be a finite set so that . Denote by the family of all finite non-empty subsets of and let and let . It is clear that and that is thick, so it suffices to show that is syndetic. We will show that Let be arbitrary. If , then , hence there exists such that . Let be such that . Therefore , hence and thus .
We now prove the converse. Assume that is a piecewise syndetic set and let be a syndetic set and be a thick set such that and let be a finite set such that . Let and, for each let . For each , choose so that . We are not going to construct sequences in with each and a sequence of infinite sets such that for any and we have . Observe that as long as we have have that and hence .
First find an infinite set and an element such that for every we have . Let . For each , let be an infinite set and be such that for each we have , and make . Finally take .
It is clear that is syndetic, since any two consecutive elements of differ by at most . Moreover, given any finite subset , say for some . Then for any with we have whenever . This shows that , finishing the proof.
— 2. Relation with topological dynamics —
A topological dynamical system (for the purposes of this post) is a pair , where is a compact Hausdorff space and is a continuous function. For instance one can take with the product topology and being the left shift (more precisely if is a point in , then ). Given a topological system and a point , the orbit closure of is the set . Orbit closures are invariant under , so for any point one can restrict to and obtain a new topological dynamical system which contains all the information about the point .
An interesting feature of topological dynamical systems are the properties of the return times to a neighborhood. Given any point and a neighborhood , what can be said of the set of return times ? It turns out that properties of the dynamical systems are often reflected in the notion of largeness possessed by the sets .
- Let and be the map . There are exactly two fixed points: and . Given any point there is a neighborhood such that the set is finite.
- Let be a compact group and let be such that the subgroup of generated by is dense in (for instance, take and to be irrational). Let be the rotation . Then for any and , the set is a Bohr set and, in particular, syndetic.
A topological dynamical system is called minimal if every point has a dense orbit, i.e., for every . It is clear that a minimal non-empty closed invariant subset of must be a minimal subsystem; therefore Zorn’s lemma guarantees that any topological dynamical system contains a minimal subsystem. It turns out that minimal systems are essentially characterized by the property that the sets of return times are syndetic.
Proof: Let , we claim that . Indeed, assume that , then for any and hence the orbit of can not be dense, contradicting the minimality. By compactness of , there exists some such that .
I claim that for any there exists and such that , this will finish the proof. Indeed, let and take . Let be such that . It follows that , which implies that .
There is a partial converse to the previous proposition, if one adds the assumption of transitivity:
Proposition 7 Let be a topological dynamical system. If there exists one point with a dense orbit and every set of the form is syndetic, then the system is minimal.
Proof: We need to show that for any we have . For this it suffices to show that , for this implies that . In order to show this we need to prove that given an arbitrary neighborhood of , the pre-orbit . Since compact Hausdorff spaces are normal, there exists a neighborhood of whose closure . The set is syndetic, so there exists a finite set such that for any we can write for some and . Therefore and hence . Since was arbitrary, we deduce that each point in the orbit of belongs to , and hence . Finally, observe that
Now let . Its indicator function is a point in the compact space . Several properties of the set are encoded in the topological dynamical system , where is the shift map. In particular, we have:
Proof: First assume that is piecewise syndetic. Using Lemma 5 we can find a syndetic set such that for any finite piece there exists a shift such that . For each let be such that . This implies that for each . Let be any limit point of the sequence , the previous observation shows that and hence is the indicator function of a syndetic set. Let be any minimal system contained in the orbit closure of , observe that . To finish the proof of this implication we need to prove that the point can not belong to .
Indeed, let and take such that . Let ; this is a neighborhood of . Assume, for the sake of a contradiction, that for some . Then for every , contradicting the fact that for some and . This contradiction ends the proof of the first implication.
Next we prove the converse: assume that such minimal subsystem exists and let . Observe that is a non-empty open subset of . Let be arbitrary and let . In view of Proposition 6, is syndetic. We will show that any finite subset of can be shifted into ; in view of Lemma 5 this will finish the proof.
Let be an arbitrary finite set and let . Observe that is an open neighborhood of . Since , there exists some such that . Unwrapping the definitions, this means that as desired.
Proposition 8 provides a way to relate a combinatorial property of a set of integers and a property of a dynamical system naturally constructed from . This relation is made more precise with the notion of sets of recurrence.
Definition 9 A set is called a set of topological recurrence if for any minimal system and any non-empty open set there exists such that .
The notion of sets of recurrence in dynamical systems is closely related to Ramsey theory on . I have mentioned sets of recurrence before in this blog, and in particular mentioned their relation with sets of chromatic recurrence. We can now add another equivalent characterization.
Theorem 10 Let . The following are equivalent:
- is a set of topological recurrence.
- is a set of chromatic recurrence (i.e., whenever there exists and such that ).
- For any syndetic set there exists such that .
- For any piecewise syndetic set there exists such that .
I will mention in passing that it is a famous question of Katzenelson whether an equivalent formulation is obtained when one replaces syndetic (or piecewise syndetic) sets by Bohr sets.
Proof: A proof of the equivalence between parts (1) and (2) was presented in my previous post. In view of Corollary 4 we have that (4) implies (2). Since finitely many shifts of a syndetic set form a finite partition of , we have that (2) implies (3), therefore it suffices to show that (1) implies (4).
Assume that (1) holds for a set , let be a piecewise syndetic and, invoking Proposition 8, let be a non-zero minimal subsystem. Let , observe that is non-empty and let be such that . Since is a non-empty open subset of , there exists such that , and in particular both and belong to , showing that .
— 3. Relation with minimal ultrafilters —
I have written before in this blog about a way to relate ultrafilters and piecewise syndetic sets; namely presenting a lovely proof of Beiglböck of a theorem of Jin (stating that whenever have positive Banach upper density, their sum is piecewise syndetic). This section will be (almost) unrelated to that proof!
There is an interesting relation between piecewise syndetic sets and certain special points in the Stone-Čech compactification of . In this previous post I presented the construction of the as the set of all ultrafilters over . An ultrafilter is a non-empty family of subsets of satisfying the following (somewhat redundant) properties:
- , but .
- If and , then .
- If then also .
- We have if and only if .
- If and we decompose into finitely many disjoint sets, then for exactly one we have .
The topology on the set of all ultrafilters over is generated by the clopen sets whenever is non-empty. With this topology, becomes a compact Hausdorff (but not metrizable) space. There is a natural action from on given by . We just created a topological dynamical system , as in the previous section this system has minimal subsystems. The following proposition characterizes piecewise syndetic sets in terms of minimal subsystems of .
Proof: First assume that belongs to a minimal subsystem of and let . This means that , and hence Proposition 6 tells us that is syndetic. I claim that any finite subset of can be shifted into a subset of , which in view of Lemma 5 will finish the proof of this implication.
Indeed, let . Unwrapping the definition we obtain for each , and hence the finite intersection is non-empty. Take any , it follows that as claimed.
Next we prove the converse direction. Assume is a piecewise syndetic set and let and be, respectively, a syndetic and a thick set such that . For each finite set , let . Since is thick, . Moreover, given any finitely many finite sets in , the intersection . Therefore the compact subsets have the finite intersection property and hence have a non-empty intersection . It is not hard to check that for any and we have ; therefore is a subsystem of . Let be a minimal subsystem of .
Let be such that . Therefore for some we have . Let be in that intersection and let . It follows that and . But since we also have that , therefore the intersection and this finishes the proof.
Observe that Corollary 4 also follows directly from this proposition.
Proposition 11 can be used to show that given any piecewise syndetic set there exists a shift (for some ) which is central (the definition of central set is explored, for instance, in this survey by Bergelson) and in my previous post (it’s Definition 4 there).