On Katznelson’s Question for skew-product systems

By Daniel Glasscock, Andreas Koutsogiannis, and Florian K. Richter

Abstract

Katznelson’s Question is a long-standing open question concerning recurrence in topological dynamics with strong historical and mathematical ties to open problems in combinatorics and harmonic analysis. In this article, we give a positive answer to Katznelson’s Question for certain towers of skew-product extensions of equicontinuous systems, including systems of the form . We describe which frequencies must be controlled for in order to ensure recurrence in such systems, and we derive combinatorial corollaries concerning the difference sets of syndetic subsets of the natural numbers.

1. Introduction

Recurrence is a central topic in the theory of dynamical systems that concerns the fundamental question of how and when a point or set recurs to its initial position. This paper addresses Katznelson’s Question, a long-standing open problem concerning recurrence in topological dynamics with strong historical and mathematical ties to open problems in combinatorics and harmonic analysis.

1.1. Main results

A topological dynamical system (henceforth, a system) is a pair , where is a compact metric space and is a continuous map. A set of positive integers is a set of recurrence for the system if there exists a point that returns arbitrarily closely to its initial position at times in , that is, . The set is a set of topological recurrence if it is a set of recurrence for all systems. Because the phase space of any system is compact, it is easy to see, for example, that is a set of recurrence. More involved examples include the set of positive differences of any infinite set and the set of squares .⁠Footnote1 Sets that are not sets of recurrence include, for example, all sets that are not divisible and all sets that are lacunary.⁠Footnote2

1

Both sets are known more generally to be sets of measurable recurrence; see Reference Fur81, Theorems 3.16 and 3.18.

2

A set is divisible if for all , there exists such that . Writing , the set is lacunary if it has “exponential growth” in the sense that .

The simplest examples of nontrivial systems are rotations of the -dimensional torus , given by , where . A set of Bohr recurrence is a set of recurrence for all finite-dimensional toral rotations. By definition, any set of topological recurrence is also a set of Bohr recurrence. Since rotations on finite-dimensional tori comprise a narrow subclass of topological dynamical systems, one is led to expect that sets of topological recurrence comprise a narrow subclass of the sets of Bohr recurrence. The extent to which this is true remains an important unsolved problem, one that was popularized in the dynamics community by Katznelson Reference Kat01. This question—and its various equivalent formulations to which we turn in a moment—is the main subject of our study.

Katznelson’s Question.

Is every set of Bohr recurrence a set of topological recurrence?

Not only is Katznelson’s Question open, there seems to be no consensus among experts as to the expected answer. There are very few concrete examples of sets which could provide a negative answer: sets which are known to be sets of Bohr recurrence but whose other dynamical recurrence properties are unknown (see Grivaux and Roginskaya Reference GR13 and Frantzikinakis and McCutcheon Reference FM12, Future Directions). The situation does not look more promising in the opposite direction: a positive answer to Katznelson’s Question is known only in a few special cases. For example, it was shown recently in Reference HKM16 that sets of Bohr recurrence are sets of recurrence for nilsystems, a class of systems of algebraic origin that generalize rotations on tori.

A natural next step toward a resolution of Katznelson’s Question is to consider skew-product extensions of equicontinuous systems. In the structure theory of topological dynamical systems initiated by Furstenberg and Veech, such systems represent a single step up in complexity from toral rotations (see Reference Gla00). The 2-torus transformation , where is continuous, is a simple example of a skew-product extension of the 1-torus rotation for which Katznelson’s Question has thus far been unresolved. Our main contribution in this paper is a positive answer to Katznelson’s Question for a class of towers of skew-product extensions over equicontinuous systems that includes this example and others.

Theorem A.

Every set of Bohr recurrence is a set of recurrence for skew-product systems of the form , where

is an equicontinuous system, and , , …, are continuous maps.

Katznelson’s Question asks whether or not recurrence along a set is guaranteed by ensuring recurrence along in all finite-dimensional toral rotations. Thus, a positive answer to Katznelson’s Question for the system begs the finer question: which rotations suffice to ensure recurrence along in the system ? In the course of our investigation, we identify the frequencies that participate in the recurrence behavior of towers of skew-product extensions of the form described in Theorem A. Surprisingly, we find that in such systems it is not enough to control for the frequencies inherent to the base equicontinuous system. This finding is in contrast with the behavior previously observed in other types of systems for which Katznelson’s Question has been answered in the affirmative, such as nilsystems Reference HKM16.

More precisely, the following theorem demonstrates that in addition to the frequencies inherent to the base equicontinuous system, it is necessary to control for new frequencies introduced by the extensions to ensure recurrence. In particular, new frequencies can be introduced even when the extensions do not increase the size of the largest equicontinuous factor of the system. For skew-product extensions of equicontinuous systems, the new frequencies introduced are the means of the skewing functions, as described in more detail in Remark 3.7.

Theorem B.

There exists an irrational toral rotation and a continuous map for which the skew-product system ,

satisfies the following:

(1)

is minimal and its largest equicontinuous factor is ;

(2)

there exists a set of recurrence for that is not a set of recurrence for .

There are several next steps suggested by Theorems A and B; we record many of them as open problems in Section 5. Questions 1 and 2 in Section 5.1 feature some simple examples of extensions of equicontinuous systems for which an answer to Katznelson’s Question is still not known. An understanding of general isometric extensions—ones which generalize skew-product extensions—from the point of view of recurrence would represent a major step toward resolving Katznelson’s Question for general distal systems. It would also direct attention toward weak mixing systems at the other end of the dynamical spectrum as the next class to analyze from this perspective.

Katznelson’s Question and its relatives were considered in equivalent, combinatorial forms long before they were popularized in dynamical terms. A subset of , respectively , is syndetic (more traditionally, relatively dense) if finitely many of its translates cover , respectively . A Bohr neighborhood of zero is a set of the form

where denotes the Euclidean distance to zero on . Bohr neighborhoods of zero and their translates are syndetic sets that generate the Bohr topology on , the coarsest topology on the integers with respect to which all trigonometric polynomials are continuous. Katznelson’s Question is equivalent to the following one, in the sense that a positive answer to one yields a positive answer to the other.

Katznelson’s Question (Combinatorial form).

If is syndetic, does the set of pairwise differences

contain a Bohr neighborhood of zero?

As with the dynamical formulation, there are only a handful of special cases in which a positive answer is known. We show in Theorem 4.9, for example, that if two translates of cover , then contains , a (periodic) Bohr neighborhood of zero.

Katznelson’s Question also finds a useful formulation in terms of -torus-valued sequences. We demonstrate the equivalence between Katznelson’s Question and the following one in Section 4.1.

Katznelson’s Question (Sequential form).

Is it true that for all and all , the set

contains a Bohr neighborhood of zero?

A sequence is Bohr almost periodic on if for all , the set

contains a Bohr neighborhood of zero. By definition, the sequential form of Katznelson’s Question has a positive answer for almost periodic sequences on ; Theorem C shows that the question has a positive answer for those sequences whose discrete derivative is almost periodic. In fact, the result applies more generally to any sequence that becomes almost periodic after finitely many discrete derivatives. It also provides a class of syndetic subsets of whose pairwise differences contain a Bohr neighborhood of zero; see Example 4.13.

Theorem C.

Let and . If the th discrete derivative of , , is Bohr almost periodic, then for all , the set

is syndetic and its set of pairwise differences, , contains a Bohr neighborhood of zero. In particular, for any such and any , the set in Equation 2 contains a Bohr neighborhood of zero.

We move next to recount the history behind Katznelson’s Question and its relatives.

1.2. History and context

A storied theorem of Steinhaus Reference Ste20 gives that the set of differences of a set of positive Lebesgue measure contains an open neighborhood of zero. Weil Reference Wei40 extended the result to locally compact groups with respect to the Haar measure. It is natural to ponder the extent to which analogues of Steinhaus’s result may hold in other settings. This becomes particularly interesting in the context of the integers, where a natural topology, the Bohr topology, is generated by all sets of the form given in Equation 1. Thus, the combinatorial form of Katznelson’s Question can be understood as an analogue to Steinhaus’s result concerning the Bohr topology on .

A more historically motivated impetus for Katznelson’s Question begins with the work of Bogolyubov Reference Bog39, who was one of the first to explore the relationship between difference sets and Bohr almost-periodic functions. In the process of giving a new proof of Bohr’s characterization of almost-periodic functions⁠Footnote3 on (as those uniformly approximable by trigonometric polynomials), he proved that if has positive upper asymptotic density, i.e.,

3

A function is Bohr almost periodic if for all , there exists such that the set has nonempty intersection with every interval in of length at least .

then the set contains a Bohr neighborhood of zero. Bogolyubov’s work seeded a vast array of generalizations to other settings, including nonabelian, nonamenable, and nondiscrete ones; Table 1 organizes many of the main results in discrete settings. To focus the narrative in this section, we will concentrate primarily on those results which have advanced our understanding in the integers.

Bogolyubov’s consideration of density ties the history of Katznelson’s Question inextricably to the history of the following related question.

Related Question.

If has positive upper asymptotic density, does its set of pairwise differences, , contain a Bohr neighborhood of zero?

Kříž Reference Kří87 gave a negative answer to the Related Question; we recount some of that history in more detail below. The historical bond between Katznelson’s Question and the Related Question is so tight that it is not possible to recount the history of one without an equal treatment of the other.

Følner Reference Føl54aReference Føl54b proved that the set contains a Bohr neighborhood of zero for any set of positive upper Banach density, i.e.,

where the supremum is over the set of left-translation invariant means (positive linear functionals of norm 1) on the bounded, real-valued functions on .⁠Footnote4 Følner also proved that when has positive upper Banach density, the set contains a Bohr neighborhood of zero up to a set of exceptions of zero Banach density. Veech Reference Vee68, Theorem 4.1, following Følner’s argument, arrived at the same conclusion when is syndetic; Veech’s argument works verbatim for sets of positive upper Banach density. The only apparent difference between Følner’s theorem and Veech’s is in their definitions of density: Følner uses the upper Weyl mean measure, while Veech considers a supremum of the values assigned by translation invariant means. We know today that these those notions of density are exactly the same (cf. footnote Footnote4).

4

It is shown in Reference Per88, Theorem 2.2a that the two quantities in Equation 3 are equal for subsets of . In , Følner’s result applies to sets of positive “upper Weyl mean measure,” which is shown in Reference BG20, Section 3 to be the same as the upper Banach density, even in more general groups and semigroups. Despite the fact that asymptotic density and Banach density are different, it can be shown that when considering the set of differences , there is no difference between assuming that has positive upper asymptotic density and assuming that has positive upper Banach density; see Reference Fur81, Theorem 3.20.

Backlinks: Reference 1, Reference 2.

Though the combinatorial form of Katznelson’s Question and the Related Question would have been natural to anyone interested in this thread of results, it seems that neither appeared explicitly in print for some time. As far as we know, the combinatorial form of Katznelson’s Question appears first in the literature as part of a more general program in Landstad Reference Lan71, p. 214:

For an amenable topological group, let be the minimal number such that is a Bohr neighbourhood whenever is a symmetric, relatively dense neighbourhood of . We have seen that in general , for abelian groups and for discrete groups. A natural question is whether this number can be reduced for some special groups.

The first explicit mention of the Related Question appears to be due to Ruzsa Reference Ruz82, p. 18.08, who attributes the question to personal communication with Flor.

It was Bochner who implicitly, if not explicitly, forged the connection between Bohr almost-periodic functions and topological dynamics; see Reference Pet89, Chapter 4 and Reference Wei00, Chapter 2 for modern accounts. In his characterization of the equicontinuous structure relation, Veech Reference Vee68 drew a connection between the combinatorial form of Katznelson’s Question and recurrence. Ellis and Keynes Reference EK72 and McMahon Reference McM78 strengthened and generalized that connection by using tools from topological dynamical structure theory to show that contains a Bohr neighborhood of zero for “many” when is syndetic. Ellis and Keynes seem to be the first to prove asymmetrical results along these lines, showing in particular that the set contains a Bohr neighborhood of zero when , , and are members of the same minimal idempotent ultrafilter. More recently, Bergelson and Ruzsa Reference BR09 showed that the triple sumset contains a Bohr neighborhood of zero when , , and are integers with and is a set of positive upper asymptotic density; stronger results were achieved in Reference LL21 under the assumption that is syndetic. Uniformity in the dimension and diameter of Bohr sets contained in triple sumsets was recently demonstrated in broad generality by Björklund and Griesmer Reference BG19.

Ruzsa Reference Ruz82Reference Ruz85 formulated both the combinatorial form of Katznelson’s Question and the Related Question and improved on Ellis and Keynes’s result by showing that contains a Bohr neighborhood of zero for many when is a set of positive upper asymptotic density. (While Reference Ruz85 was never published, several of the results appear in Reference HR16.) Ruzsa also expounded on a theorem of Kříž Reference Kří87 that answers the Related Question in the negative: there exists a set of positive upper asymptotic density whose set of differences do not contain a Bohr neighborhood of zero. This result was recently strengthened by Griesmer Reference Gri21, answering a question in Reference GR09, p. 196: there exists a set of positive upper asymptotic density whose set of differences does not contain a translate of any Bohr neighborhood of zero.

The first more recent mention of the combinatorial form of Katznelson’s Question in print is found in Glasner Reference Gla98, who connected the problem to fixed points of actions of minimally almost-periodic groups. He shows that for a negative answer to Katznelson’s Question, it suffices to construct a minimally almost-periodic Polish monothetic topological group that acts with no fixed points by homeomorphisms on a compact space. For a collection of related problems, see Pestov Reference Pes07.

Katznelson Reference Kat01 was perhaps the first to explicitly formulate the eponymous question as one about recurrence in topological dynamics and is credited for popularizing this question in the dynamics community. Bergelson, Furstenberg, and Weiss Reference BFW06 employed tools and techniques from ergodic theory to prove, among other results, an asymmetric result reminiscent of Følner’s: if have positive upper Banach density, then contains the intersection of a translate of a Bohr neighborhood of zero with a set containing arbitrarily long intervals. Griesmer Reference Gri12 improved on this result by weakening the positive Banach density assumption on one of the sets. Boshernitzan and Glasner Reference BG09 summarized what is known about Katznelson’s Question and other related questions in the framework of dynamics and recurrence, and Huang, Shao, and Ye Reference HSY16 formulated higher-order analogues of Katznelson’s Question in the framework of nilsystems and nil-Bohr sets.

Some of the most recent progress on the Katznelson’s Question was made by Host, Kra, and Maass, who gave a positive answer for nilsystems and their proximal extensions (see Reference HKM16, Proposition 3.8 and Theorem 4.1). Nilsystems are translations of compact homogeneous spaces of nilpotent Lie groups; Host, Kra, and Maass showed that in a minimal nilsystem, any set of recurrence for the largest equicontinuous factor is a set of recurrence for the nilsystem. They also showed that if is a proximal extension⁠Footnote5 of minimal systems, then every set of recurrence for is a set of recurrence for .

5

An extension is proximal if for all with and all , there exists such that .

Backlinks: Reference 1, Reference 2.

Host, Kra, and Maass’s results combine with ours to give a list of systems in which a positive answer to Katznelson’s Question is known: nilsystems; skew-product extensions of equicontinuous systems by 1-tori; systems which support a measure with respect to which the transformation exhibits mixing on the -orthocomplement of the Kronecker factor; and inverse limits, proximal extensions, and factors of such systems. Beyond a few other sporadic examples, to our knowledge, this is a complete list.

1.3. Outline of the article

The article is organized as follows. In Section 2, we lay out the notation, terminology, and results from topological dynamics required for our main theorems. Section 3 covers basic results about skew products; Theorems A and B are proved in Sections 3.2 and 3.4, respectively. In Section 4, we elaborate on Katznelson’s Question and its consequences in a combinatorial setting, including a proof of Theorem C in Section 4.2. We end the paper with Section 5 by discussing a number of open questions and directions.

2. Notation, terminology, and prerequisites

We denote the set of integers and positive integers by and , respectively. The (additive) 1-torus is denoted by and is equipped with the metric induced by the function that measures the Euclidean distance to the nearest integer. Throughout, for convenience, Cartesian products of metric spaces are equipped with the (taxicab) metric.

2.1. Combinatorics and topological dynamics

For and , define

As defined in the introduction, a (topological dynamical) system is a pair consisting of a compact metric space and a continuous map . A system is minimal if for all , the set is dense in .

In this paper, we will focus on the recurrence of points in systems. The following definition helps to make this precise. (The interested reader can consult Reference HKM16, Theorem 2.3 and Reference BG09, Theorems 5.3 and 5.6 for a number of other equivalent characterizations of sets of topological recurrence, including the one mentioned in the first paragraph of Section 1.1.)

Definition 2.1.

The set of -returns of a system is

A set is a set of topological recurrence if for all systems and all ,

Lemma 2.2.

Let be a system. For all and ,

Proof.

Let . The conclusion of the lemma follows by noting that for all , both of the conditions and are equivalent to the existence of such that .

2.2. Bohr sets, almost periodicity, and equicontinuity

Bohr sets, which play a central role in Katznelson’s Question and in this paper, are closely related to the topics of almost periodicity and equicontinuity. In this section, we define Bohr sets and collect the prerequisite results necessary for the proofs of main theorems.

Definition 2.3.

A set is a Bohr set if it contains the positive elements of a Bohr neighborhood of zero, that is, if there exists , , and such that

The set is a Bohr set if there exists such that is a Bohr set. A subset of is a Bohr set (also, a set of Bohr recurrence) if it has nonempty intersection with all Bohr subsets of .

Remark 2.4.

The family of Bohr subsets of is a filter: it is upward closed and closed under intersections. Dually, the family of Bohr subsets of is partition regular: at least one cell of any finite partition of a Bohr set is a Bohr set. A set is Bohr if and only if it has nonempty intersection with all Bohr sets.⁠Footnote6 Also, note that is a Bohr set if and only if for all and , . This helps to explain why such sets are called sets of Bohr recurrence; see also the terminology in Definition 2.1.

6

That every Bohr set has nonempty intersection with every Bohr set follows by definition. Conversely, suppose that has the property that for all Bohr sets . It follows that is not a Bohr set. By the definition of Bohr sets, there exists a Bohr set such that , whereby . Supersets of Bohr sets are Bohr sets, so is a Bohr set.

Remark 2.5.

The completion of with the Bohr topology—the topology generated by Bohr neighborhoods of zero and their translates—yields its Bohr compactification, . Addition on induces a binary operation on that makes it a compact (nonmetrizable) abelian group. In this context, a set is a Bohr set if and only if it contains the preimage (under the canonical injection of into ) of an open neighborhood of in , and a set is a Bohr set if and only if is an accumulation point of the image of in . We mention the Bohr compactification here only to help motivate the terminology; we do not have any use for particulars concerning in this paper, so we do not develop the details any further.

Lemma 2.6.

If is a Bohr set, then for all , the sets and are Bohr sets.

Proof.

Let and be such that

For , let

It is easy to check that

The result follows by the relation , as both and are Bohr sets, and by the fact that .

Let be a compact abelian group, and let be addition by a fixed element . The map is an isometry (with respect to a translation-invariant metric on ), and the set of times at which a point visits a nonempty open set is a Bohr set. In fact, the same conclusion can be reached under the weaker, topological assumption that the family of maps is equicontinuous; see Lemma 2.10 below and the remark following it.

Definition 2.7.

A system is equicontinuous if the family of maps is equicontinuous, i.e., for all , there exists such that for all with and all , .

Equicontinuity is closely related to the dynamical phenomenon of almost periodicity, defined next. See Lemma 2.9 for the connection which is most relevant to this work.

Definition 2.8.

Let be a metric space, and let . The sequence is (Bohr) almost periodic on if for all , the set of -almost periods

is a Bohr set. Replacing all instances of with yields the definition of a Bohr almost-periodic function on as defined in the introduction. The mean of a real-valued almost-periodic sequence is the quantity .

The following is a collection of useful classical results relating Bohr sets, almost periodicity, and equicontinuity; see Reference Pet89, Chapter 4 for a modern presentation of the ideas.

Lemma 2.9.

Let . The following are equivalent:

(1)

the sequence is almost periodic;

(2)

there exist an equicontinuous system , a point , and a continuous function such that for all , .

Moreover, the same statement holds with replaced by and with “equicontinuous system” replaced by “minimal equicontinuous system” in condition .

It is a well-known consequence of equicontinuity, at least in minimal systems, that the set of return times of a point to a neighborhood of itself is a Bohr set, but we were unable to find a convenient reference in the literature concerning nonminimal systems. The argument is short so we provide it here.

Lemma 2.10.

Let be an equicontinuous system. For all , the set

is a Bohr set.

Proof.

First we will show that for all and , the set

is a Bohr set. Let be continuous, equal to 1 at , and equal to 0 outside of an open ball of radius about . It follows from Lemma 2.9 that the sequence is almost periodic. Since , the set of -almost periods of , a Bohr set, is contained in .

Now we will prove the statement in the lemma. Let . Let be sufficiently small so that for all with and for all , . Let be a -dense subset of . By the previous paragraph, the set

where is defined as in Equation 5, is a Bohr set since it is the intersection of finitely many Bohr sets.

We will show that is a subset of the set in Equation 4. Let and . There exists such that . Since , , and , we have by the triangle inequality that , as was to be shown.

In fact, Bohr sets can be used to characterize equicontinuous systems: a minimal system is equicontinuous if and only if for all and , the set is a Bohr set. We do not have need for this fact, so we omit the proof.

2.3. Dynamical forms of Katznelson’s Question

Katznelson’s Question can be stated in several different equivalent forms. In this section, we describe two dynamical forms; some of its combinatorial forms are presented in Section 4.1. For our purposes, it will be most convenient to phrase Katznelson’s Question in terms of the size of the set of -returns of a system.

Definition 2.11.

A system has Bohr large returns if for all , the set of -returns is a Bohr set.

Katznelson’s Question and the following one are equivalent, in the sense that one has a positive answer if and only if the other does. It is this formulation of Katznelson’s Question that we will address in the next section.

Question D1.

Do all topological dynamical systems have Bohr large returns?

That Katznelson’s Question and Question D1 are equivalent follows from the marginally finer fact that a system has Bohr large returns if and only if sets of Bohr recurrence are sets of recurrence for . Indeed, suppose has Bohr large returns, and let be a set of Bohr recurrence. As explained in Remark 2.4, the set is a Bohr set. For all , the set is a Bohr set, whereby . Since was arbitrary, the set is a set of recurrence for . Conversely, suppose that a system does not have Bohr large returns: there exists such that is not a Bohr set. As explained in Remark 2.4, the set is a Bohr set, a set of Bohr recurrence, that is not a set of recurrence for .

Question D2.

If is a minimal topological dynamical system, is it true that for all nonempty, open , the set

is a Bohr set?

Questions D1 and D2 are equivalent. Indeed, that a positive answer to Question D2 implies one for D1 follows from the fact that any system contains a minimal subsystem , and . On the other hand, a positive answer to Question D1 combines with the following lemma to immediately give a positive answer to Question D2.

Lemma 2.12.

Let be a minimal system. For all nonempty, open , there exists such that

Proof.

The second containment is immediate: if , then , whereby . To see the first, let be such that contains a non-empty open set and its -neighborhood. Since is minimal, there exists such that for all , there exists such that . Let be such that for all with and for all , . Now, if , there exists such that . It follows that there exists such that and . Therefore, , whereby .

3. Recurrence and hidden frequencies in skew-product systems

In this section, we prove Theorems A and B. The first gives a positive answer to Katznelson’s Question for certain towers of skew-product extensions by -tori over equicontinuous systems, while the second demonstrates that skew-product extensions can introduce new “frequencies” that must be controlled to ensure recurrence.

3.1. Skew-product dynamical systems

We collect here the basic notation and terminology for skew-product systems, winding numbers, and lifts of torus-valued maps.

Definition 3.1.

Let be a system, and let be a continuous map. The skew-product system is defined by where

For , define by and

so that .

We will frequently consider real-valued skewing functions ; the skew-product system in this case is defined by implicitly composing the map with the quotient map .

Definition 3.2.

Let be continuous. There exists a continuous map with the property that . The winding number of is equal to ; it is an integer that can be shown to be independent of . If has winding number equal to zero, then descends to a continuous function satisfying . We refer to as the continuous lift of to .

Remark 3.3.

The winding number of a continuous function counts the number of times the function “wraps around” the circle. The winding number of a sum of functions is the sum of their winding numbers. For , the winding number of is easily seen to be equal to the winding number of . It follows that the winding number of the function , defined in Definition 3.1, is times the winding number of .

3.2. Returns in towers over equicontinuous systems: a proof of Theorem A

In this section, we prove Theorem A using the reformulation of Katznelson’s Question described in Section 2.3. At the heart of Theorem A is a simple idea that is quickly illustrated in the case of a single skew product by the 1-torus over a rotation on the 1-torus.

Special case of Theorem A.

For all and all continuous , the skew-product system ,

has Bohr large returns.

Proof.

Let and be continuous. Let . In order to show that , we must demonstrate the existence of a point for which and . (Recall that all Cartesian products in this work are equipped with the metric.)

Let . If has nonzero winding number, then so does , and it follows by the intermediate value theorem that there exists such that . It follows that .

If, on the other hand, the function has zero winding number, then it has a continuous lift . Put . For any , the mean value theorem for integrals gives the existence of a point such that . This implies that .

In either case, we find that contains a Bohr set, whereby has Bohr large returns.

To prove Theorem A, we improve on this idea in two ways. First, we replace the base -toral rotation by a general equicontinuous system. If is totally disconnected—as it is when is an odometer, for example—the argument can no longer appeal to winding numbers or the intermediate value theorem. The fact that the result continues to hold for not-necessarily-connected base systems shows that it has less to do with connectedness and, as we will see, more to do with the fact that the real numbers are well ordered. Second, to extend the result to certain towers of skew-product extensions, we iterate the argument, using the fact that entire fibers exhibit recurrence.

The first step in the proof of Theorem A is to show that partial sums of real-valued, almost-periodic sequences are close to their mean along a Bohr set.

Proposition 3.4.

Let be almost periodic, and let be its mean. For all , there exists a Bohr set such that for all , there exists such that

Proof.

Let . Let be the set of -almost periods for . The set is a Bohr set that we will show satisfies the conclusions of the proposition.

Let . Define by

and note that has mean . Because is an -almost period for , the sequence takes -steps,” in the sense that for all , . Since has mean and it takes -steps, there exists for which , as was to be shown.

The following theorem proves Theorem A in the case of a single skew-product extension over an equicontinuous system.

Theorem 3.5.

Let be an equicontinuous system, and let be continuous. The skew-product system has Bohr large returns.

Proof.

Because is equicontinuous, Lemma 2.10 gives that the set

is a Bohr set.

We consider two cases. In case 1, for all , the point 0 is in the image of the map , while in case 2, there exists for which 0 is not in the image of the map .

Suppose we are in case 1. To see that has Bohr large returns, let . We claim that . Let . Since 0 is in the image of the map , there exists such that . It follows that , whereby , as was to be shown.

Suppose we are in case 2 so that there exist for which . We claim that we can assume that . Indeed, to prove that has the Bohr large returns, it suffices by Lemma 2.2 to prove that the system has Bohr large returns. Define . Note that (following the notation established in Definition 3.1), so that . Since is equicontinuous and is continuous, we can proceed by replacing by , by , and under the assumption that is not in the image of the map .

Let be the quotient map from to , let be a section of that is continuous at all points of except 0, and define . Since is not in the image of , the map is continuous. Moreover, by construction. Fix and define by . The system is equicontinuous, so by Lemma 2.9 the sequence is almost periodic.

Let ; our aim is to show that is a Bohr set. Let be the mean of , and let be the Bohr set from Proposition 3.4 (with as ). Define

Note that is a Bohr set. We will show that .

Let . It follows by Proposition 3.4 and the fact that that there exists such that

Define . Note that . It follows by the definition of and Equation 8 that

Since , we have additionally that . Combining these facts, we see that

It follows that , as was to be shown.

In the following theorem, we establish the inductive step for an iterative procedure that allows us to handle the multiple skew-product extensions that appear in Theorem A.

Theorem 3.6.

Let be a system, and let be continuous. If the skew-product system has Bohr large returns, then for all continuous , the skew-product system ) defined by

has Bohr large returns.

Proof.

Let be continuous. If has nonzero winding number, put . If has winding number equal to zero, let be a continuous lift of to , and put .

Let . It follows from our assumptions that the set

is a Bohr set. We will show that .

Let . To show that , we will show that there exists such that

Note that

where the third coordinate function, denoted by , depends on and is defined by

For all , the winding number of is times the winding number of , and, in the case that has winding number equal to zero, the function , defined by replacing with in Equation 10, is a continuous lift of to with .

Since , there exists such that

Since commutes with rotation in the second coordinate, it follows, in fact, that Equation 11 holds for all .

If has nonzero winding number, then so does ; it follows by the intermediate value theorem that there exists such that . On the other hand, if has winding number equal to zero, then the mean value theorem for integrals combines with the fact that is a continuous lift of to with mean to guarantee the existence of a such that . Since , it follows that .

In either case, it follows from Equation 11 that for all , the point satisfies Equation 9, as was to be shown.

Combining Theorems 3.5 and 3.6, we can prove Theorem A.

Proof of Theorem A.

According to the reformulation of Katznelson’s Question in Question D1 in Section 2.3, we need to prove that the systems described in the statement of Theorem A have Bohr large returns. That fact follows by a simple induction argument, appealing to Theorem 3.5 for the base case and Theorem 3.6 for the inductive step.

Remark 3.7.

The proofs of Theorems 3.5 and 3.6 tell us which frequencies it suffices to control to ensure recurrence in a tower of skew-product extensions of the type described in Theorem A. For those skewing functions with zero winding number, we must control for the average of a continuous lift of ; those skewing functions with nonzero winding number do not introduce any additional frequencies. The example in Theorem B shows that controlling for the averages of the skewing functions with zero winding number is indeed necessary for recurrence. In the case of more general isometric extensions of equicontinuous systems, we do not know how to identify frequencies beyond those in the equicontinuous factor that influence recurrence.

Corollary 3.8.

Sets of Bohr recurrence are sets of recurrence for factors, proximal extensions (cf. footnote Footnote5), and inverse limits of the types of skew-product tower systems described in the statement of Theorem A.

Proof.

It is easy to check that factors of systems and inverse limits of families of systems that have Bohr large returns also have Bohr large returns. It is a consequence of Reference HKM16, Proposition 3.8 that proximal extensions of systems with Bohr large returns have Bohr large returns. Thus, the statement in question is an immediate corollary of Theorem A and the equivalences between the different forms of Katznelson’s Question described in Section 2.3.

3.3. Skew-product extensions by the 1-torus

The two main results in this section concern skew-product extensions of equicontinuous systems by the 1-torus. Theorem 3.9 concerns general equicontinuous systems; the result will be useful in the proof of Theorem C but may also be of independent interest. Theorem 3.10 is a complement to the classic theorem of Gottschalk and Hedlund; we elaborate on this in Remark 3.11 after the proof.

Theorem 3.9.

Let be a minimal, equicontinuous system, and let be continuous. If the skew-product system is not minimal, then it is equicontinuous.

Proof.

Fix , and let . Making use of the family of automorphisms of described by , , there exists a closed subgroup of such that ; the details are left to the reader as an exercise. Since is minimal, the system is minimal if and only if .

Suppose that the system is not minimal so that ; there exists such that . We will show that the system is equicontinuous. Using the family of second-coordinate rotation automorphisms, it suffices to show that the system is equicontinuous.

Let ; it is a 1-torus with metric induced by , the Euclidean distance to zero. Let be the quotient map, and consider the skew-product system , where we endow with the metric. Let . Following the same reasoning as above, by the definition of , for all , .

Let be such that for all , ; that is, the graph of is equal to . It follows that

Since is closed, the map is continuous. Note that in the system , we have for all that , whereby for all .

We will leverage the fact that is discrete to show that the system is equicontinuous. Recall that .

Claim 1 (Local lifts).

There exists such that for any ball of diameter at most , there exists a continuous map such that (that is, is a continuous lift of and such that for all , .

Proof.

Since is continuous and is compact, there exists some such that for all with ,

Let be any ball of diameter centered at a point . Let be any point in for which , and consider the interval

The projection map restricted to , which we denote by , is continuous and surjective. Moreover, it is injective because the length of is smaller than and points that have the same image under are at least apart. Therefore the map is a continuous bijection between compact spaces, which implies that it is a homeomorphism. Let denote its inverse. Since and in light of Equation 12, we have . This ensures that the map is a well-defined continuous function from to satisfying for all . Since we clearly have , the proof is complete.

Let be an -dense subset of . For each , define and appeal to Claim 1 to find , a continuous lift of . Note that for all for which is nonempty, the function

Indeed, since both and are lifts of , we have . Therefore, the function maps into . By Claim 1, the function differs between two points on by at most . Since any two points of are separated in distance by at least , the conclusion follows.

Let be such that ; such an exists because is equicontinuous, and this will be fixed for the rest of the proof. For , define , and note that , so that is defined and is continuous on . Define by

that takes values in can be seen by applying and using the fact that . Since , , and are all continuous on , the function is continuous. We claim that if are such that , then . Indeed, for , there exists by Equation 13 a value such that and . It follows that

Since is -dense, we have that . Therefore, by Equation 14, we can define by defining, for each , the function to be equal to on . Since each is continuous, the function is continuous.

Summarizing the previous paragraph, there exists a continuous function (that depends on ) such that for all and all ,

Since is uniformly continuous and is discrete, there exists such that for all with , .

To show that is equicontinuous, it suffices to show that the system is equicontinuous. Let . We will show that there exists such that for all with , for all ,

This will show, by definition, that the system is equicontinuous.

Define .

Claim 2 (Equicontinuity constant).

There exists such that for all with and for all , there exists such that and

Proof.

Since is finite and each is uniformly continuous on the closure , there exists such that for all and all with ,

By the equicontinuity of the family , there exists such that for all and all with , we have

Let with and . As is -dense, there exists such that . We see then that

whereby , .

For the second conclusion in Claim 2, using the fact that , we have

and

so , .

Since , the inequality in Equation 18 implies that

Since , , using Equation 17, we have that

as was to be shown.

Let be the equicontinuity constant from Claim 2. Let be such that . We will show that for all ,

implies that

Note that Equation 19 holds when since . Thus, the inequality in Equation 16 follows from a simple induction on and the fact that .

Suppose that and that Equation 19 holds. Since , it follows by Claim 2 that there exists such that , . Since , there exist such that and . It follows from Claim 1 (note that ) and Equation 19 (note that ) that

whereby .

Now we compute, using Equation 15,

The same equalities hold with and replaced by and , respectively. Since , . Therefore, by Claim 2,

verifying Equation 20 and finishing the proof of the theorem.

The Kronecker factor of a system is its largest equicontinuous factor. In the next theorem, we prove a general result concerning minimality and the Kronecker factor of skew products on the 2-torus. We will need this result to verify property 1 in Theorem B. The supremum norm on is denoted by .

Theorem 3.10.

Let be an irrational rotation. Let be continuous, and let . If the sequence , where , is unbounded, then the skew-product system is minimal and has Kronecker factor .

Proof.

Let such that . First, we will show that the system is minimal. Suppose for a contradiction that it is not. By Theorem 3.9, the system is equicontinuous, hence Lemma 2.10 gives that the set

is a Bohr set, and hence is syndetic. Fix . For all , the fact that implies that . Thus, there exists a function such that for all , . Since is continuous, the function must be constant: for all , . Since has mean , we get that , which implies that . Using the fact that , it is easy to show that since the sequence is bounded along a syndetic subsequence, it is bounded. This is in contradiction to our assumption, concluding the proof that the system is minimal.

Now we will show that the system has Kronecker factor . Since is minimal and distal, its Kronecker factor is determined by the regional proximal relation Reference Vee68, Theorem 1.1: is regionally proximal to if and only if for all , there exists with and such that and . Because the factor is equicontinuous, if and are regionally proximal, then .

Because commutes with rotation in the second coordinate of , to prove that the system has Kronecker factor , it suffices to prove that for all , the points and are regionally proximal. Let , and let . Let be sufficiently small so that if , then . Because is minimal, the set of return times of to the -neighborhood of the point is syndetic; therefore, the set

is syndetic. Since is syndetic, it follows from our assumptions that the sequence is unbounded along .

Let be such that is -dense in , and choose such that . We will show that there exists such that and . Since , we will have that , and since was arbitrary, this will finish the proof that and are regionally proximal.

Since and the mean of is , there exist such that and . By our choice of , for all ,

By our choice of , there exist such that are both within of . By repeatedly appealing to Equation 21, we have that , from which it follows that . Similarly, . Since is continuous, by the intermediate value theorem, the image of restricted to an -ball about is all of . Therefore, there exists , , such that . It follows that the point satisfies and , as was to be shown.

Remark 3.11.

Theorem 3.10 is a complement to the classic theorem of Gottschalk and Hedlund Reference GH55, Theorem 4.11, which asserts that the sequence is bounded if and only if there exists a continuous function such that . In this case, the -homeomorphism demonstrates the topological conjugacy between the skew-product system and the rotation . Therefore, if the sequence is bounded, the skew-product system is equicontinuous, and it is minimal if and only if , , and are linearly independent over the rationals.

3.4. The hidden frequencies example: a proof of Theorem B

In this section, we prove Theorem B by giving an example of a minimal skew-product system on in which recurrence in the Kronecker factor (the system’s largest equicontinuous factor, the base rotation) does not suffice for recurrence in the system. Such an example stands in sharp contrast to other systems in which the answer to Katznelson’s Question is known, and it demonstrates at least some of the difficulty of answering the question for more general systems. We use the notation of skew-product systems from Definition 3.1.

Define by

Since , there exists so that , where is the quotient map. The system in the proof of Theorem B will be a skew-product system of the form ,

for certain . Recall from Definition 3.1 that in the system , the map is implicitly precomposed with the quotient map .

Lemma 3.12.

For all , for all ,

Proof.

Let . Since the map is -periodic on , it suffices to verify Equation 22 with replaced by for all real values of . By the Faulhaber formulae for for , we have that

Skipping the algebra, the sum of interest is equal to

The claim follows since, for every ,

The following theorem gives a sufficient condition for a skew-product system of the form on to be minimal. It is quick to check that the function defined above satisifies the hypotheses of the theorem; in fact, this is precisely how was chosen.

Theorem 3.13 (Reference HL89, Theorem 1.4, combined with the remark following it).

Suppose that , , and satisfy

is -times continuously differentiable on ;

;

for all , ;

; and

for all , , where is the th partial quotient of (as in Equation 24) and is the denominator of the th convergent of .

Defining such that , the skew-product system on is minimal.

The following lemma explains the choice of in the system we construct; its second conclusion follows from Remark 3.11 and Theorem 3.13.

Lemma 3.14.

There exists and a sequence for which:

(1)

the nearest integer to is coprime to and ; and

(2)

the sequence is unbounded.

Proof.

Let be a sufficiently rapidly increasing sequence such that, on defining , , and , we have for all that . (Take, for example, .) Let be the real number whose sequence of simple continued fraction partial quotients is :

We claim that this and the sequence satisfy the conclusion of the lemma.

Denote by the th continued fraction convergent of . The following are standard facts in the theory of continued fractions Reference Khi63, Chapter 1: (where is as defined in the previous paragraph); and are coprime; and

in particular, the nearest integer to is . This shows that the condition in 1 is satisfied.

According to Theorem 3.13 (with ), the skew-product system is minimal. It follows from Remark 3.11 (where ) and the fact that the rotation is not minimal that the sequence must be unbounded, as was to be shown.

Lemma 3.15.

For any sequence , there exists such that for infinitely many , .

Proof.

Define . Let be the Lebesgue measure on . Since for all , it follows from Fatou’s lemma that the set

has measure at least . Any irrational in this set satisfies the conclusion of the lemma.

Proof of Theorem B.

Let and be as guaranteed by Lemma 3.14. Appealing to Lemma 3.15, let and pass to a subsequence of so that for all , . Let be a Lipschitz constant for , and let . Passing to a further subsequence of , we may assume that for all , and .

It follows from Lemma 3.14 and Theorem 3.10 that the system is minimal and has Kronecker factor . Put . Since , the set is a set of recurrence for . We have only left to verify that is not a set of recurrence for . It suffices to show that for all and all , .

Let and , and let be the nearest integer to so that . Since and are coprime, there exists a permutation of such that for all , . We estimate

It follows from Equation 22 that

Since , we have that , as was to be shown. This concludes the proof of Theorem B.

4. Katznelson’s Question in a combinatorial framework

As recounted in Section 1.2, combinatorial forms of Katznelson’s Question and its relatives were considered long before Katznelson and others popularized them in dynamical form. In this section, we provide some of those combinatorial formulations and prove the equivalence between them. We also prove Theorem C, a combinatorial corollary to our main dynamical result, Theorem A.

Recall that a set is syndetic if there exists a finite set such that . The set is piecewise syndetic if there exists a finite set such that contains arbitrarily long intervals (i.e., is thick).

4.1. Combinatorial forms of Katznelson’s Question

In what follows, the phrase “Question A implies Question B” means that a positive answer to Question A yields a positive answer to Question B. We say that Questions A and B are equivalent if implies and implies . We will prove the equivalence between the various forms of Katznelson’s Question posed in Section 1.1 indirectly, beginning first with some alternate combinatorial formulations.

Remark 4.1.

For most of the questions posed in this paper, there is not a material difference between the set of positive integers and the set of all integers . Some combination of the following three facts generally suffices to prove the equivalence between analogous questions in these two settings:

(1)

a Bohr neighborhood of zero is symmetric about zero and, when restricted to , is a Bohr set;

(2)

if is a Bohr set, then the set contains a Bohr neighborhood of zero; and

(3)

if is a syndetic subset of , then is syndetic in , and if is syndetic in , then is syndetic in .

Since syndetic sets are piecewise syndetic, a positive answer to the following question implies a positive answer to the combinatorial form of Katznelson’s Question.

Question C1.

If is piecewise syndetic, does its set of differences contain a Bohr neighborhood of zero?

To see that Question C1 implies the combinatorial form of Katznelson’s Question, we will use the fact that if is piecewise syndetic, then it is broken syndetic: there exists a syndetic set with the property that for all finite , there exists such that ; see Reference Ruz85, Proof of Theorem 1. It follows immediately that , and hence that being a Bohr implies that is a Bohr set.

While the property of being syndetic is not partition regular, the related notion of piecewise syndeticity is; see Reference Fur81, Theorem 1.24. Since one cell of any finite partition of is piecewise syndetic, Question C1 implies the following useful combinatorial form of Katznelson’s Question.

Question C2.

If is a finite partition of , does the set contain a Bohr neighborhood of zero?

To see that Question C2 implies C1, it suffices to show that Question C2 implies Katznelson’s Question. If is syndetic, then there exists such that . It follows from C2 that the set

is a Bohr set, yielding a positive answer to Katznelson’s Question. Thus, Questions C1 and C2 are equivalent forms of Katznelson’s Question. This equivalence is also proved using different terminology in Reference Ruz85, Theorem 1.

Remark 4.2.

The foregoing sequence of questions may lead one to wonder whether or not the family of subsets of that have Bohr large differences is partition regular. This is not the case, as can be seen by the following example. Let be a set of positive upper asymptotic density that does not have Bohr large differences; such a set exists by an example of Kříž Reference Kří87. The set of differences of is syndetic Reference Fur81, Proposition 3.19, so there exists such that . Put . Because , the set has Bohr large differences, but no cell of the partition of has Bohr large differences.

Lemma 4.3.

Katznelson’s Question is equivalent to its combinatorial formulation.

Proof.

We will show the equivalence between Question D1 (from Section 2.3) and Question C2. This equivalence has been documented a number of times in the literature; see, for example, Reference BG09, Lemma 4.5 or Reference Kat01, Proof of Theorem 2.1. Since the argument is short, we provide it here for completeness.

To see that Question C2 implies Question D1, suppose is a system and . Let be a cover of by finitely many balls of diameter less than . Fix , and pull the cover of back through the map to a cover so that implies that . It is quick to check that , whereby a positive answer to Question C2 implies a positive answer to Question D1.

That Question D1 implies C2 relies on a correspondence principle. Suppose , and let be such that for all , . The sequence belongs to the compact metric space , on which we consider the left-shift map . Put . Let be such that if satisfy , then . If , then there exists such that . Since has a dense orbit in , there exists such that for , …, . It follows that , whereby and are in the same element of the cover. Therefore, . We’ve shown that , whereby a positive answer to Question D1 implies a positive answer to Question C2.

In Reference Kat01, Katznelson’s Question appears in terms of chromatic numbers of certain graphs on . For , define a graph on by putting an edge between if and only if . Denote by the chromatic number of the graph .

Question C3 (Reference Kat01).

If is a Bohr set, is ?

This question is quickly seen to be a reformulation of Question C2, and hence of Katznelson’s Question. Indeed, a finite partition is exactly a finite coloring of . By Remark 2.4, the set is a Bohr set if and only if for all Bohr sets ,

Note that Equation 25 holds if and only if there are adjacent vertices in the graph with the same color. Thus, both Questions C2 and C3 ask whether or not Equation 25 holds for all finite colorings of and all Bohr sets .

The following question appears easier to answer than the combinatorial form of Katznelson’s Question; it is, in fact, shown to be equivalent by an elementary argument. Note that it was shown in Reference EK72 that the set contains a Bohr set when is syndetic.

Question C4.

If is syndetic, contains , and satisfies , is the triple sumset a Bohr set?

To see that Katznelson’s Question implies Question C4, suppose has the properties stipulated in Question C4. Since , if the combinatorial form of Katznelson’s Question has a positive answer, then is a Bohr set. To see that Question C4 implies Katznelson’s Question, we borrow a clever argument from Reference Ruz85, Theorem 2; see also Reference HR16, Proof of Theorem 2.1. Suppose the combinatorial form of Katznelson’s Question has a negative answer: There exists a syndetic set for which is not a Bohr set. Put

Clearly, is syndetic, contains 0, and satisfies . Considering residues modulo 4, it is quick to show that

Since is not a Bohr set, neither is the set . By Lemma 2.6, it follows that the set is not a Bohr set, answering Question C4 in the negative.

There is a useful dialogue between sequences (more generally, functions) and dynamics; see Reference BG09, Theorem 5.6 for a particular connection between sequences and recurrence and Reference Wei00 for a broader view. This is part of the reason why the sequential formulation of Katznelson’s Question echoes the dynamical one.

Lemma 4.4.

Katznelson’s Question is equivalent to its sequential formulation.

Proof.

It is easier to see the equivalence between the sequential form of Katznelson’s Question and Question C2. To see that the former implies the latter, suppose ; without loss of generality, we may assume that the sets are disjoint. Choose distinct points on the 1-torus, , …, , and define by and, for , and . If , then

Thus, a positive answer to the sequential form of Katznelson’s Question implies a positive answer to Question C2.

To show the converse, let and . Cover by finitely many balls of diameter : . Pull this cover of back through to get a finite partition , and restrict this partition to one of . It is quick to check that

whereby a positive answer to Question C2 yields a positive answer to the sequential form of Katznelson’s Question.

Remark 4.5.

It is clear from the argument that the converse implication in Lemma 4.4 depends only on the total boundedness of . Thus, one can formulate an equivalent question that appears more difficult to answer by replacing by an arbitrary, totally bounded metric space in the sequential formulation of Katznelson’s Question.

Remark 4.6.

The following special case of the sequential form of Katznelson’s Question is, in fact, equivalent to the more general form stated in the introduction: Is it true that for all and all , the sequence is such that the set

contains a Bohr neighborhood of zero? To see that the two are equivalent, it suffices by Lemma 4.4 to show that a positive answer to the special case yields a positive answer to Question C2.

Suppose , and define so that for all , . Choose so that , and define

Note that if and , then . It follows by the special case of the sequential form of Katznelson’s Question stated above and by Lemma 2.6 that the set

is a Bohr set. Note that if is such that , then there exists such that . Thus, for all , there exists such that . This implies that the set , which is a Bohr set by Lemma 2.6, is contained in the set , yielding a positive answer to Question C2.

Remark 4.7.

A minimal system has Bohr large returns if and only if for all continuous and all , the observable sequence is such that

is a Bohr set. We will not have use for this connection explicitly in this paper, so we leave the verification of this fact to the curious reader.

We conclude this section with a formulation of Katznelson’s Question in terms of the lengths of zero-sum blocks of cyclic-group-valued sequences. A zero-sum block for is an interval on which sums to zero; this notion does not appear to be well-studied, but did appear recently in the literature Reference CHM19.

Question C5.

If , is the set of lengths of zero-sum blocks

a Bohr set?

To see that Question C5 implies Question C2, suppose . Define by and, for , and . Put , and define by modulo . If , then , whereby for some . Therefore, the set of lengths of zero-sum blocks for is contained in the set , implying that a positive answer to Question C5 yields a positive answer to Question C2.

To see the converse, let . Define by . For , let be the set of those for which . The set is a subset of the set in Equation 26. Since is a Bohr set, so is the set in Equation 26.

Remark 4.8.

Finite cyclic groups are not essential to the formulation of Question C5. If is a compact abelian group with invariant metric and , an -sum block is an interval on which sums to within of the identity 0. Using the same reasoning as above, it is easy to see that Question C5 is equivalent to the ostensibly more difficult question obtained by replacing the set in Equation 26 with

It should also be noted that a positive answer to the sequential form of Katznelson’s Question for a class of sequences does not necessarily imply a positive answer to Question C5 for the same class. For example, the sequential form of Katznelson’s Question trivially has a positive answer for almost-periodic sequences. It does not follow, however, from the equivalence described above that Question C5 has a positive answer for almost-periodic sequences. In fact, it is true that Question C5 has a positive answer for almost-periodic sequences, but this is the result of Theorem C which requires additional arguments.

A number of open questions closely related to Katznelson’s Question are presented in Section 5.2.

4.2. Combinatorial results: a proof of Theorem C

In this section, we provide positive answers to the combinatorial form of Katznelson’s Question for 2-syndetic sets and for certain classes of syndetic sets that arise naturally in topological dynamics. The first is accomplished by a simple combinatorial argument, while the second—made precise in the statement of Theorem C—is derived from Theorem A.

Theorem 4.9.

For all -colorings , either or there exists for which . In particular, the set is a Bohr set.

Proof.

If , then the conclusion of the theorem holds. Otherwise, there exists . Since , we see that . Similarly, . It follows that and , and hence that .

The next corollary follows immediately from Theorem 4.9. It is still not known whether or not the final conclusion in Theorem 4.9 holds for all 3-colorings of ; see Question 3 in Section 5.1.

Corollary 4.10.

Let . If , then there exists such that . In particular, the set is a Bohr set.

We turn now to the proof of Theorem C. We will make use of systems of the form , where the map is defined by

Thus, the system is an iterated skew-product system, where the identity skewing map is repeated many times, over a base skew-product system . We denote by the projection onto the th coordinate.

For , define . A straightforward induction shows that for all ,

where we define, for , , and, for ,