Variations on Topological Recurrence

Recurrence properties of systems and associated sets of integers that suffice for recurrence are classical objects in topological dynamics. We describe relations between recurrence in different sorts of systems, study ways to formulate finite versions of recurrence, and describe connections to combinatorial problems. In particular, we show that sets of Bohr recurrence (meaning sets of recurrence for rotations) suffice for recurrence in nilsystems. Additionally, we prove an extension of this property for multiple recurrence in affine systems. 1. Topological recurrence Van der Waerden’s classic theorem [37] states that any finite coloring of the integers contains arbitrarily long monochromatic progressions. This has led to numerous refinements and strengthenings, with some of these obtained via the deep connections to topological dynamics introduced with the proof of Furstenberg and Weiss [19]. A direction that has been studied extensively is what restrictions can be placed on the step in the arithmetic progression, and in dynamics this corresponds to what sets arise as sets of recurrence. Recurrence properties of systems and the associated sets of recurrence are classical notions both in topological dynamics and in additive combinatorics, and have numerous classically equivalent characterizations. Part of this article is a review of these connections, many of which are scattered throughout the literature, and we point out numerous open questions. Part of this article is new, particularly connections to objects that have recently shown to play a role in topological dynamics, such as nilsystems. For both single and multiple recurrence, the class of nilsystems (see Section 4 for definitions) plays a natural role. This is reflected in work in the ergodic context on multiple convergence along arithmetic progressions [21]. In the topological context, a higher order regionally proximal relation was introduced in [23], where the connection to nilsystems was made. Further deep connections between these notions and that of topological recurrence were made in [25]. Nilsystems have also been used to construct explicit examples of sets of multiple recurrence, for example in the work of [14, 25]. Thus the relation between recurrence and its connections with nilsystems have become a natural direction for further study. The second author was partially supported by NSF grant 1200971 and the third author was partially supported by the Bézout Chair of the Université Paris-Est Marne-la-Vallée. 1 2 BERNARD HOST, BRYNA KRA, AND ALEJANDRO MAASS Our main focus is how to formulate finite versions of recurrence related to van der Waerden’s Theorem. One way is to fix a length for the progressions and then characterize the sets of recurrence for this fixed length. We then study classifying dynamical systems by their recurrence properties along arithmetic progressions of this length, seeking necessary or sufficient conditions for such recurrence. In various guises, this problem has been studied by dynamicists and we consider this point of view in Section 2. In particular, we study a question asked by Katznelson [26]: if R is a set of recurrence for all rotations, is it a set of recurrence for all minimal topological dynamical systems? (See Section 3 for the definitions.) We give a partial answer to this question, showing that it holds when one restricts to the class of nilsystems (Theorem 4.1) and its almost proximal extensions. We then turn to the similar questions for multiple recurrence. In this setting, we show (Theorem 5.13) that if R is a set of s-recurrence for s-step affine nilsystems, then it is also a set of t-recurrence for all t ≥ s for the same class of systems. A summary of these implications is given in Figure 1. A second way to finitize van der Waerden’s Theorem is by fixing the number of colors and studying the associated sets of recurrence. This point of view has largely been ignored by dynamicists and we take this approach in Section 7, where we mainly pose further directions for study. Throughout this article, we assume that (X,T ) denotes a (topological dynamical) system, meaning that X is a compact metric space and T : X → X is a homeomorphism. While our primary focus is on topological recurrence, there are also measure theoretic analogs, where the underlying space is a probability measure preserving system (X,B, μ) endowed with a measurable, measure preserving transformation T : X → X. Combinatorially, this corresponds to Szemerédi’s Theorem and the connection to ergodic theory has been well studied. While the measure theoretic and topological settings give rise to similar results, there are some differences and we point out some of the known measure theoretic analogs and pose some related questions. 2. Variations on recurrence 2.1. Single recurrence. Throughout, we focus on minimal systems (X,T ), meaning that no proper closed subset of X is T -invariant. Definition 2.1. We say that R ⊂ N is a set of (topological) recurrence if for every minimal system (X,T ) and every nonempty open set U ⊂ X, there exists n ∈ R such that U ∩ T−nU 6= ∅. Notation. If x ∈ X and U ⊂ X is an open set, we write N(x, U) = {n ∈ N : Tx ∈ U} for the return times of the point x to the neighborhood U and N(U) = {n ∈ N : U ∩ T−nU 6= ∅} VARIATIONS ON TOPOLOGICAL RECURRENCE 3 for the return times of the set U to itself. In case of ambiguity, we include the transformation in our notation and write NT (x, U) or NT (U). Thus R ⊂ N is a set of recurrence if for every minimal system (X,T ) and every nonempty open set U ⊂ X, there exists n ∈ R such that n ∈ N(U). We recall a standard definition: Definition 2.2. A subset of integers is syndetic if the differerence between two consecutive elements is bounded. We have the following classical equivalences (see, for example [19, 18, 17, 3, 4, 30]). We omit the proofs, as simple recurrence is a special case of the more general result for multiple recurrence (Theorem 2.5): Theorem 2.3. For a set R ⊂ N, the following are equivalent: (i) R is a set of recurrence. (ii) For every system (X,T ) and every open cover U = (U1, . . . , Ur) of X, there exists j ∈ {1, . . . , r} and n ∈ R such that n ∈ N(Uj). (iii) For every finite partition N = C1 ∪ · · · ∪ Cr of N, there is some cell Cj containing two integers whose difference belongs to R. (iv) Every syndetic subset E of N contains two elements whose difference belongs to R. (v) For every system (X,T ) and every ε > 0, there exist x ∈ X and n ∈ R such that d(Tnx, x) < ε. (vi) For every system (X,T ), there exists x ∈ X such that inf n∈R d(Tx, x) = 0. (vii) For every minimal system (X,T ) there exists a dense Gδ set X0 ⊂ X such that for every x ∈ X0, inf n∈R d(Tx, x) = 0. A set R satisfying characterization (iv) is referred to as (chromatically) intersective in the combinatorics literature. It is easy to check that the existence of some n ∈ R satisfying any of properties (i), (ii) or (v) implies that there exist infinitely many n ∈ R with the same property. Example 2.4. For S ⊂ N, write S−S = {s′−s : s, s′ ∈ S, s′ > s}. Furstenberg [18] showed that if S is infinite, then S − S is a set of recurrence and this follows immediately from characterization (iii) in Theorem 2.3. More generally, it is easy to check that if for every n ∈ N there exists Sn ⊂ N such that |Sn| = n and Sn − Sn ⊂ R, then R is a set of recurrence. We defer further examples of sets of recurrence until we have defined the more general notion of multiple recurrence. There is another equivalent formulation of recurrence due to Katznelson [26]. For a set R ⊂ N, the Cayley graph GR is defined to be the graph 4 BERNARD HOST, BRYNA KRA, AND ALEJANDRO MAASS whose vertices are the natural numbers N and whose edges are the pairs {(m,m + n) : m ∈ N, n ∈ R}. The chromatic number χ(R) is defined to be the smallest number of colors needed to color GR such that any two vertices connected by an edge have distinct colors. Katznelson showed that characterization (iii) of Theorem 2.3 for a set of recurrence R is equivalent to the associated Cayley graph GR having infinite chromatic number. For the analogous notion of a set of measure theoretic recurrence, where the underlying space is a probability measure space and the transformation is a measurable, measure preserving transformation, we have a similar list of equivalent characterizations, where a finite partition of N is replaced by sets of positive upper density. As every minimal system (X,T ) admits a T -invariant measure with full support, a set of measurable recurrence is also a set of topological recurrence. However, an intricate example of Kriz [27] shows that the converse does not hold. 2.2. Multiple recurrence. Most of the formulations of single recurrence generalize to multiple recurrence: Notation. For ` ≥ 1, we write N (U) = {n ∈ N : U ∩ T−nU ∩ T−2nU ∩ · · · ∩ T−`nU 6= ∅} for the return times of the set U to itself along a progression of length `+ 1. In case of ambiguity, we include the transformation in our notation and write N ` T (U). Theorem 2.5. Let ` ≥ 1 be an integer. For a set R ⊂ N, the following properties are equivalent: (i) For every minimal system (X,T ) and every nonempty open set U ⊂ X, there exists n ∈ R such that n ∈ N `(U). (ii) For every system (X,T ) and every open cover U = (U1, . . . , Ur) of X, there exists j ∈ {1, . . . , r} and n ∈ R such that n ∈ N (Uj). (iii) For every finite partition N = C1 ∪ . . . ∪ Cr of N, there is some cell Cj that contains an arithmetic progression of length `+ 1 whose common difference belongs to R. (iv) Every syndetic set E ⊂ N contains an arithmetic progression of length `+ 1 whose common difference belongs to R. (v) For every system (X,T ) and every ε > 0, there exist x ∈ X and n ∈ R such that sup 1≤j≤` d(T x, x) < ε. (vi) For every system (X,T ), there exists x ∈ X such that inf n∈R sup 1≤j≤` d(T x, x) = 0. VARIATIONS ON TOPOLOGICAL RECURRENCE 5 (vii) For every minimal system (X,T ), there exists a dense Gδ-set X0 ⊂ X such that for every x ∈ X0, inf n∈R sup 1≤j≤` d(T x, x) = 0. Definition 2.6. A set satisfying any of the equivalent properties in Theorem 2.5 is called a set of `-recurrence; in particular, a set of 1-recurrence is a set of recurrence. A set of `-recurrence for all ` ≥ 1 is a called a set of multiple recurrence. When we want to emphasize that we are discussing single recurrence, instead of just writing a set of recurrence, we say a set of single or simple recurrence. The proofs of these equivalences are well known and appear scattered in the literature (see, for example [19, 18, 17, 3, 4, 29, 30, 12, 7, 15, 9]) and so we only include brief sketches of the proofs. Proof. (i) =⇒ (vii) For ε > 0, define Ωε to be {x ∈ X : there exists n ∈ R such that d(Tx, x) < ε, . . . d(T x, x) < ε}. Then Ωε is an open subset ofX. Let U ⊂ X be an open ball of radius δ < ε/2. By hypothesis, there exists n ∈ R such that U ∩ T−nU ∩ · · · ∩ T−`nU 6= ∅. This intersection is included in Ωε and so Ωε is dense in X. Then X0 = ⋂ m∈N Ω1/m is a Gδ set that satisfies the statement. (vii) =⇒ (vi) This is immediate by applying (vii) to a minimal closed invariant subset of X. (vi) =⇒ (v) Obvious. (v) =⇒ (ii) Let ε be the Lebesgue number of the cover U , meaning that any open ball of radius ε is contained in some element of this cover. Let x ∈ X and n ∈ R be associated to ε as in (v). Let j ∈ {1, . . . , r} be such that the ball of radius ε around x is included in Uj . Then all of the points x, Tnx, . . . , T `nx belong to this ball and thus to Uj . (ii) =⇒ (iii). This is a standard application of the topological version of Furstenberg’s Correspondence Principle. Given the partition N = C1 ∪ · · · ∪ Cr, there exist a system (X,T ), a partition X = U1 ∪ · · · ∪ Ur of X into clopen sets, and a point x ∈ X such that for every n ∈ N, we have Tnx ∈ Uj if and only if n ∈ Cj . (iii) =⇒ (iv) Choose r ∈ N such that (E− 1)∪ (E− 2)∪ . . .∪ (E− r) ⊃ N and then chose a partition N = C1 ∪ · · · ∪ Cr such that Cj ⊂ E − j for j ∈ {1, . . . , r}. (iv) =⇒ (i) Choose x ∈ X and set E = {n : Tnx ∈ U}. As for single recurrence, the existence of some n ∈ R satisfying any of properties (i), (ii), or (v) immediately implies the existence of infinitely many n ∈ R with the same property. It is easy to verify that a set of (single or multiple) recurrence must satisfy several necessary conditions: it must contain infinitely many multiples of 6 BERNARD HOST, BRYNA KRA, AND ALEJANDRO MAASS every positive integer (consider the powers of the transformation) and it can not be lacunary (by constructing an irrational rotation that fails to recur). Furthermore, the family of sets of recurrence has the Ramsey property (see Section 6). The classic theorem of van der Waerden shows that N is a set of multiple recurrence. Furstenberg [18, Theorem 2.16] shows that N(x, U) is a set of multiple recurrence for any open set U and point x ∈ U . This is also a particular case of a more general theorem of Huang, Song, and Ye [25], reviewed in Theorem 5.8. There are many other known examples of sets of multiple recurrence: any IP-set (a set which contains all the finite sums of an infinite set of integers, see Definition 3.9), the set {p(n) : n ∈ N}, where p(n) is any nonconstant polynomial with p(0) = 0, the shifted primes {p − 1: p is prime} and {p + 1: p is prime}, as well as other examples in the literature (see for example [19, 36, 6, 4, 5, 13]) There are also examples in the literature that show that sets of multiple recurrence are different than sets of single recurrence. For example, Furstenberg [18] gives an example of a set of single recurrence that is not a set of double recurrence and Frantzikinakis, Lesigne and Wierdl [14] give examples of sets of `-recurrence that are not sets of (` + 1)-recurrence. We give a more general characterization of such sets in Section 5.2. We note that all of the examples constructed in this way are large, in the sense that they have positive density. However, there are characterizations of single recurrence for which we do not have a multiple analog: Question 2.7. Is there an equivalent characterization of multiple recurrence analogous to Katznelson’s characterization in terms of the chromatic number of an associated graph? For example, is being a set of multiple recurrence equivalent to infinite chromatic number for some associated hypergraph? Along similar lines, we do not know of a simple construction, like that of the difference set, that suffices to produce multiple recurrence: Question 2.8. Is there a sufficient condition, analogous to that given in Example 2.4, that suffices for being a set of multiple recurrence? 2.3. Simultaneous recurrence. More generally, we can study recurrence for commuting transformations and not just powers of a single transformation: Definition 2.9. The set R ⊂ N is a set of `-simultaneous recurrence if for any compact metric space X endowed with ` commuting homeomorphisms T1, . . . , T` : X → X such that the system (X,T1, . . . , T`) is minimal and any nonempty open set U ⊂ X, there exists n ∈ R such that U ∩ T−n 1 U ∩ . . . ∩ T −n ` U 6= ∅. VARIATIONS ON TOPOLOGICAL RECURRENCE 7 A set of `-simultaneous recurrence for all ` ≥ 1 is a called a set of simultaneous recurrence. Taking T1 = T, T2 = T 2, . . . , T` = T `, it is obvious that any set of simultaneous recurrence is also a set of multiple recurrence. We do not know if the converse holds: Question 2.10. Does there exist a set of multiple recurrence that is not a set of simultaneous recurrence? All of the examples of sets of multiple recurrence given in Section 2.2 are also known to be sets of simultaneous recurrence. All parts of Theorem 2.5 have natural analogs for simultaneous recurrence. To ease the notations, we restrict ourselves to ` = 2. It is easy to check that the analog of condition (iii) holds: namely, R is a set of recurrence if for every partition N = C1 ∪ . . .∪Cr, there exists x, y ∈ N and n ∈ R such that (x, y), (x+n, y), (x, y+n) all lie in the same cell Cj for some j ∈ {1, . . . , r}. One can give similar formulations for the other equivalences in Theorem 2.5 for simultaneous recurrence. Unsurprisingly, we do not know how to address the analogs of Questions 2.7 and 2.8 for simultaneous recurrence. 2.4. Pointwise recurrence. Definition 2.11. A set R ⊂ N is a set of pointwise recurrence if for every minimal system and every x ∈ X, inf n∈R d(Tx, x) = 0. The analog for multiple pointwise recurrence is not defined, as one can construct an example (such as using symbolic dynamics) of a minimal system (X,T ), as open set U ⊂ X, and x ∈ U such that N2(x, U) = ∅. In particular, N is not a set of pointwise multiple recurrence. However, in a minimal system, there is always a dense set of points that are multiply recurrent. Recall that by characterization (vii) of Theorem 2.3, if R is a set of recurrence then this property holds for x in a dense Gδ of X. Comparing the definition of pointwise recurrence with characterization (vi) of recurrence in Theorem 2.3 makes this property seem natural. However, being a set of pointwise recurrence turns out to be a significantly stronger assumption. Sárkőzy [33] (using number theoretic methods) and Furstenberg [18] (using dynamics) showed that the set of squares is a set of recurrence, but Pavlov [32] showed that it is not a set of pointwise recurrence. Similarly, it follows from results in Pavlov that if one takes S to be a sufficiently fast growing sequence, then S − S is not a set of pointwise recurrence (but as noted in Example 2.4, it is a set of recurrence). Notation. For t ∈ R, we use ‖t‖ to denote the distance of t to the nearest integer. For t ∈ T = R/Z, ‖t‖ denotes the distance to 0. 8 BERNARD HOST, BRYNA KRA, AND ALEJANDRO MAASS Example 2.12. One can check directly that for every α ∈ T, the set R = {n ∈ N : ‖n2α‖ ≥ 1/4} is not a set of pointwise recurrence by using an affine nilsystem (see Example 5.11). In [14], the authors show, in particular, that R is a set of measurable recurrence, and thus also of recurrence. We briefly outline their method. If α is irrational, by Weyl equidistribution, for every non-zero t ∈ [0, 1), the averages