Some open problems on multiple ergodic averages
Contents
 1 Introduction

2 Some useful tools and observations
 2.1 Characteristic factors
 2.2 GowersHostKra seminorms
 2.3 The factors and their structure
 2.4 A general strategy
 2.5 The polynomial exhaustion technique
 2.6 Equidistribution of polynomial sequences on nilmanifolds
 2.7 Pleasant and magic extensions
 2.8 Furstenberg correspondence principle
 2.9 Equivalent problems for sequences
 3 Some useful notions
 4 Problems related to general sequences
 5 Problems related to polynomial sequences
 6 Problems related to sequences arising from smooth functions
 7 Problems related to random sequences
 8 Extended bibliography organized by theme
1. Introduction
1.1. What is this?
This is a first attempt to organize a list of some open problems on three closely related topics:

The limiting behavior of single and multiple ergodic averages.

Single and multiple recurrence properties of measure preserving systems.

Universal patterns, meaning, patterns that can be found in every set of integers with positive upper density, and related problems on higher dimensions.
The list of problems is greatly influenced by my personal interests and is by no means meant to be a comprehensive list of open problems in the area widely known as ergodic Ramsey theory. Almost exclusively, problems related to actions of commuting measure preserving transformations are considered, and even within this confined class there are a few important topics not touched upon (for instance, the richness of the return times in various multiple recurrence results). For material and a list of problems that goes beyond the scope of this set of notes we refer the reader to the survey articles [24, 27, 28] and the references therein.
Whenever appropriate, I include the “original source” and a short history of the problem, as well as a hopefully accurate and up to date list of related work already done. I plan to update these notes from time to time, so you are welcome to contact me and help me improve them.
1.2. The general theme
A very general framework that can be used to describe the bulk of the problems listed below is the following: We are given a measure space with , invertible measure preserving transformations , bounded measurable functions , and sequences . In some cases we also consider sequences that depend on several integer variables, but let’s stick to the single variable case for the time being.
The first family of problems concerns the study of the limiting behavior of the so called multiple ergodic averages
(1) 
where and . One would like to know whether these averages converge as (in or pointwise), find some structured factors that control their limiting behavior (called characteristic factors), and if possible, find a formula, or a usable way to extract information, for the limiting function. When , such problems have been studied extensively and in several cases solved (see the survey paper [170] for a variety of related results). Our main concern here is to study the averages (1) when . To get manageable problems, one typically restricts the class of eligible sequences, usually to be polynomial sequences, sequences arising from smooth functions, sequences related to the prime numbers, or random sequences, and also assumes that the transformations commute, or, to get started, that they are all equal. On the other hand, because of the nature of the implications in combinatorics that we are frequently interested in, it is not desirable to assume anything about the particular structure of each individual measure preserving transformation. Typically, the tools used to attack such problems include elementary uniformity estimates, ergodic structure theorems (like the one in [115]), and equidistribution results on nilmanifolds. More on that on subsequent sections.
The second family of problems concerns the study of expressions of the form
(2) 
where has positive measure. One wants to know whether such expressions are positive for some , or even better, for lots of (for instance on the average), and if possible, get some explicit lower bound that depends only on the measure of the set and on (optimally this is going to be of the form ). Such multiple recurrence results are typically obtained by carrying out an in depth analysis of the limiting behavior of the averages (1). Usually they are not hard if an explicit formula of the limiting function is known, but they can be very tricky in the absence of such a formula, even when we work with very special systems of algebraic nature.
Concerning the third family of problems, and restricting ourselves to subsets of , one is interested to know, for example, whether every set of integers with positive upper density^{1}^{1}1The upper density of a set is defined by . contains patterns of the form
for some (or lots of) and . A typical instance is the celebrated theorem of Szemerédi [177] stating that every set of integers with positive upper density contains arbitrarily long arithmetic progressions (this corresponds to the case for , ). Using a correspondence principle of H. Furstenberg, one can translate such statements to multiple recurrence statements in ergodic theory; an equivalent problem is then to show that the expressions (2) are positive for some when all the measure preserving transformations are equal. Similar questions can be asked on higher dimensions, concerning patterns that can be found on subsets of with positive upper density. Such questions correspond to multiple recurrence statements when the transformations commute. This approach was originally used by H. Furstenberg in [91] to give an alternate proof of Szemerédi’s theorem using ergodic theory. Subsequently, H. Furstenberg and Y. Katznelson gave the first proof of the multidimensional Szemerédi theorem [97] and the density HalesJewett theorem [100], and V. Bergelson and A. Leibman proved the polynomial extension of Szemerédi’s theorem [35] (currently no proof that avoids ergodic theory is known for this result). And the story does not end there, in the last two decades new powerful tools in ergodic theory were developed and used, and are currently being used, to prove several other deep results in density Ramsey theory. The reader will find several such applications in subsequent sections and the extended bibliography section. Several additional applications can be found in the survey articles and the references therein [24, 27, 28].
Recently, an additional motivation for studying such problems has surfaced. It has to do with potential implications in number theory, in particular a connection to problems of finding patterns in the set of the prime numbers. Knowing that every set of integers with positive upper density contains patterns of a certain sort could be an important first step towards proving an analogous result for the set of primes. This idea originates from work of B. Green and T. Tao [104], where it was used to show that the primes contain arbitrarily long arithmetic progressions. It was also subsequently used by T. Tao and T. Ziegler [183] to show that the primes contain arbitrarily long polynomial progressions.
1.3. General conventions and notation
We are going to use the following notation: , , , if is a measure preserving system then , . We use the symbol when some expression is majorized by a constant multiple of some other expression.
Acknowledgment. I would like to thank B. Kra and E. Lesigne for helpful comments.
2. Some useful tools and observations
2.1. Characteristic factors
A notion that underlies the study of the limiting behavior of several multiple ergodic averages is that of the characteristic factor(s). Implicit use of this notion was already made on the foundational article of H. Furstenberg [91], but the term “characteristic factor” was coined in a paper of H. Furstenberg and B. Weiss [101].
Given a probability space and a collection of measure preserving transformations , we say that the subalgebras of are characteristic factors for the averages
(3) 
if the following two conditions hold:

is invariant for ,

whenever , we have where for .^{2}^{2}2Equivalently, if for some , then .
If in addition one has , then we call this common subalgebra a characteristic factor for the averages (3).
2.2. GowersHostKra seminorms
When analyzing the limiting behavior of the averages (3), an intermediate goal is to choose characteristic factors that are as simple as possible, and typically simple for us means that the corresponding factor systems have very special algebraic structure. Very often this step is carried out by controlling the norm of the averages (3) by the GowersHostKra seminorms. Similar seminorms were first introduced in combinatorics by T. Gowers [103] and their ergodic variant (that is more relevant for our study) was introduced by B. Host and B. Kra [115]. For an ergodic system and function , they are defined as follows:
It is shown in [115] that for every the limit above exists, and . For nonergodic systems the seminorms can be similarly defined, the only difference is that is the ergodic decomposition of the measure with respect to . If further clarification is needed, we write ^{3}^{3}3A measure preserving system is weak mixing if the product system is ergodic, or equivalently, if for every with one has ., then . for every . We remark that if a measure preserving system is weak mixing , or , where is defined by , thus defined, is a seminorm on
2.3. The factors and their structure
The seminorms invariant subalgebras that satisfy induce
(4) 
As a consequence, if for some one is able to produce an estimate of the form
(5) 
then one knows that the factors are characteristic for mean convergence of the averages (3). Under such circumstances, one gets characteristic factors with the soughtafter algebraic structure. This is a consequence of a result of B. Host and B. Kra [115] stating that for ergodic systems the factor system is an inverse limit of step nilsystems^{5}^{5}5A step nilmanifold is a homogeneous space where is a step nilpotent Lie group, and is a discrete cocompact subgroup of . A step nilsystem is a system of the form where is a step nilmanifold, , is defined by , is the normalized Haar measure on , and is the completion of the Borel algebra of .. Depending on the problem, it may be more useful to think of the previous structure theorem as a decomposition result; for every ergodic system and , for every and , there exist measurable functions with norm at most , such that

;

; and ;

is a step nilsequence^{6}^{6}6A step nilsequence is a uniform limit of sequences of the form where is a step nilmanifold, , , and is Riemann integrable on . for almost every .
Such a decomposition also holds for nonergodic systems (see Proposition in [62]).
Combining the hypothetical seminorm estimates (5) with the aforementioned structure theorem (or the decomposition result), the problem of analyzing the limiting behavior of the averages (3) is reduced to a new problem that amounts to proving certain equidistribution properties of sequences on nilmanifolds. Tools for handling such equidistribution problems have been developed in recent years, thus making such a reduction very much worthwhile. Some examples of equidistribution results of this type can be found in [8, 78, 106, 107, 140, 141, 152, 153].
2.4. A general strategy
Summarizing, when one is against a multiple recurrence problem in ergodic theory, or more generally any problem that can be solved by analyzing the limiting behavior of the multiple ergodic averages (3), very often a useful approach is to try to work out the following three steps:^{7}^{7}7This rough plan is already implicit in the foundational paper of Furstenberg [91], the only difference is that in [91] the role of nilsystems played the much larger class of distal systems. Depending on the problem, this approach may offer some advantages, but for several recent applications it appears that the class of distal systems is just too broad to deal with directly.

Produce seminorm estimates like the ones in (5).

Use a structure theorem or a decomposition result to reduce matters to nilsystems.

Use qualitative or quantitative equidistribution results on nilmanifolds to end the proof.
The reader can find several examples demonstrating this approach, or variants of it, to prove multiple recurrence and convergence results, as well as related applications in combinatorics, in the following articles: [7, 33, 40, 41, 58, 61, 62, 64, 76, 77, 79, 82, 83, 84, 85, 87, 90, 101, 108, 114, 115, 116, 119, 120, 124, 142, 154, 168, 173, 191, 195, 196]. Depending on the problem, the difficulty of each step varies; typically the first step is elementary and is carried out by successive uses of the CauchySchwarz inequality and an estimate of van der Corput^{8}^{8}8This states that if are complex numbers bounded by , then for every integer between and we have (or Hilbert space variants of it), the second step involves the use of (an often minor) modification of the structure theorem of B. Host and B. Kra, and the third step is a combination of algebraic and analytic techniques.
2.5. The polynomial exhaustion technique
We explain a technique that is often used to produce seminorm estimates of the type (5). It is based on an induction scheme (often called PET induction) introduced by V. Bergelson in [22]. Let be a family of real valued sequences, and suppose that one wishes to establish seminorm estimates of the form
(6) 
where
Variations of this method could also be used to get similar estimates for some multiple ergodic averages involving commuting transformations.
2.5.1. The method
The main idea is to use some variation of van der Corput’s fundamental estimate and bound the left hand side in (6) by an expression that involves families of sequences of smaller “complexity”. Our goal is after a finite number of iterations to get families of sequences that are simple enough to handle directly. The details depend on the family of sequences at hand, but typically, after the first iteration, we get an upper bound by an average over of the norm of multiple ergodic averages with iterates taken from the family of sequences
(7) 
where is fixed (so in particular independent of ) and chosen so that the family has smaller “complexity” than except possibly for a finite number of .
To be able to carry out this plan one first needs to take care of some elementary, but often not so easy, preparatory steps:

Define a suitable collection of families of sequences for which the desired seminorm estimates are easy to obtain directly.

Define a suitable collection of families of sequences that contains and the family .

Define a notion of equivalence and then a partial order in so that: every decreasing sequence , with , is eventually constant, and if , then there exists such that and for all but finitely many ( is defined as in (7)).
These conditions guarantee that there exists and an appropriate choice of sequences , such that the iteration
(not to be confused with an exact sequence!) produces families that belong to for a set that is big enough for our purposes. Practically, this means that after applying van der Corput’s estimate a finite number of times, we have good chances to be able to bound the left hand side in (6) by a much simpler expression for which we can prove the desired seminorm estimates directly.
This strategy has been employed successfully in several instances and produced seminorm estimates of the form (6) for linear sequences [115], polynomial sequences [116, 142], block polynomials of fixed degree [90], and some sequences coming from smooth functions of polynomial growth [31, 79]. Notice a common desirable feature that these sequences share: after taking successive differences (meaning iterating the operation ) a finite number of times we arrive to sequences that are either constant or piecewise asymptotically constant. This feature is not shared by several other sequences worth studying, for example, random sequences of integers, the sequence of primes, and the sequences , . In such cases one has to modify the PET induction approach or abandon it altogether and try something different.^{9}^{9}9For instance, for the first two sequences, it turns out to be more effective to utilize the random features of the sequences at hand, and get seminorm estimates by comparing the corresponding multiple ergodic averages with other deterministic ones that are better understood.
2.5.2. An example
Let be a family of essentially distinct polynomials, meaning, all polynomials and their pairwise differences are not constant. Then one can define as the collection of all families consisting of a single linear polynomial, for such families establishing an estimate of the form (6) is easy, and as the collection of all families of essentially distinct polynomials.
The tricky part is to define a partial order in that satisfies the third requirement mentioned in Section 2.5.1. This is done as follows: First, define the degree of a family of nonconstant polynomials to be the maximum of the degrees of the polynomials in the family. Next, let be the subfamily of polynomials of degree in , and let denote the number of distinct leading coefficients that appear in the family . The vector is going to be the complexity of the family . We identify two families that have the same complexity and we order the set of all possible complexities lexicographically, meaning, if and only if in the first instance where the two vectors disagree the coordinate of the first vector is greater than the coordinate of the second vector. We order the (equivalence classes) of families of polynomials accordingly. One easily verifies that every decreasing sequence of complexities is eventually constant, and if , then there exists such that the family is in and has complexity strictly smaller than that of for all but finitely many .
Using this strategy, seminorm estimates similar to those in (6) were established [116] for all essentially distinct polynomials except for a few cases (one has to dig into the details to see why this argument misses some cases) that were handled in [142] (for alternate proofs see Lemma 4.7 in [79] or Theorem 1.4 in [61]).
2.6. Equidistribution of polynomial sequences on nilmanifolds
Let be a nilmanifold, and , and be sequences. In several of the applications we have in mind one is at some point called to prove that the sequence defined by ,^{10}^{10}10Such sequences cover as special cases sequences of the form , defined on the product of the nilmanifolds . To see this let , , , , where denotes the identity element of the group . is equidistributed on some subset of .^{11}^{11}11If is a nilmanifold we say that a sequence is equidistributed in a subnilmanifold (suppose that for ) of if for every one has where denotes the Haar measure on .
Often is , or a subnilmanifold of , and in a few cases a union of subnilmanifolds of . Such problems are typically much easier to handle when , since in this case one can utilize Weyl’s equidistribution theorem^{12}^{12}12This states that a sequence is equidistributed on a subnilmanifold of a torus (suppose that for ) if and only if for every nontrivial character one has . in order to reduce matters to estimating certain exponential sums. Unfortunately, such a convenient reduction is not available for all nilmanifolds, and checking equidistribution in this broader setup can be very challenging even for simple sequences.^{13}^{13}13The use of representation theory has not proven to be of much help in this case. Luckily, the situation is much better understood when all the sequences are given by integer polynomials, in this case we call the sequence polynomial; next, we are going to state some key results.
In the forthcoming discussion we assume that is a connected nilmanifold. By we denote the connected component of the identity element in ,^{14}^{14}14For technical reasons we assume that is simply connected and that . When is connected, we can always arrange so that has these additional properties. and we let and be the natural projection. It is important to notice that has much simpler structure than . Indeed, if is connected, then is a connected compact Abelian Lie group, hence, a torus (meaning for some ), and as a consequence every nilrotation in is (isomorphic to) a rotation on some torus. In general, the nilmanifold may be more complicated, but it is the case that every nilrotation in is (isomorphic to) a unipotent affine transformation on some torus^{15}^{15}15A map is said to be affine if for some homomorphism of and . The homomorphism is said to be unipotent if there exists so that . In this case we say that the affine transformation is a unipotent affine transformation. (see Proposition 3.1 in [83]). Iterates of such transformations can be computed explicitly^{16}^{16}16If is a unipotent affine transformation, and , then one can easily check that the coordinates of the sequence are polynomial in ., so one is much more comfortable to be dealing with equidistribution problems that involve unipotent affine transformations on some torus than with general niltransformations.
The following qualitative equidistribution results were established by A. Leibman in [140]:

A polynomial sequence is always equidistributed in a finite union of subnilmanifolds of .

A polynomial sequence is equidistributed in if and only if the sequence is equidistributed in .
The second statement gives a very effective way for checking equidistribution of polynomial sequences. We illustrate this with a simple example. Suppose that is an ergodic nilrotation (meaning the transformation is ergodic) and we want to show that the polynomial sequence is equidistributed in for every . In the case where is connected the nilmanifold is a torus, therefore, according to the previous criterion, it suffices to show that if is an ergodic element of (this is the case if the coordinates of are rationally independent), then for every the sequence is equidistributed in . This is a well known fact, and can be easily verified using Weyl’s equidistribution theorem and van der Corput’s fundamental estimate. If is not necessarily connected, one needs to show that if is an ergodic unipotent affine transformation, then the sequence is equidistributed for every .^{17}^{17}17In some cases, for instance, when one seeks to prove equidistribution of a sequence in some unspecified nilmanifold, one can use a lifting argument in order to reduce matters to the case where is connected. Such a simplification does not seem to be possible for the example just presented (the lifting does not preserve the ergodicity assumption). Although this is somewhat harder to establish, it follows again by Weyl’s equidistribution theorem modulo some straightforward computations.
In other cases^{18}^{18}18For instance, when one seeks to study equidistribution properties of the sequence , or tries to prove uniform convergence for the sequence to the integral of , where is an ergodic element and . one needs quantitative variants of the previous qualitative equidistribution results. Such a result was proved by B. Green and T. Tao [106]; we are not going to give the precise statement here because this would require to introduce too much additional notation.
2.7. Pleasant and magic extensions
Motivated by the work of T. Tao [181], H. Towsner [186], T. Austin [9], and B. Host [111], introduced new tools that help us handle some multiple ergodic averages. In particular, a key conceptual breakthrough that first appeared in [9], is that in some instances by working with suitable extensions of a family of commuting measure preserving systems (called “pleasant extensions” in [9] and “magic extensions” in [111]), characteristic factors of the corresponding multiple ergodic averages may be chosen to have particularly simple structure, a structure that is not visible when one works with the original systems (the idea of passing to an extension in order to simplify some convergence problems already appears in [101]). This is a rather counterintuitive statement since characteristic factors of extensions are extensions of characteristic factors of the original systems, so we going to explain a simple instance where such an approach works. Suppose that one wants to prove mean convergence for the averages
where are commuting measure preserving transformations acting on the probability space and . Although an estimate that relates the norm of these averages with the GowersHostKra seminorms of the individual functions with respect to either or is not feasible, the following estimate is valid
where
(it is shown in [111] that
where and denotes the algebra of invariant sets.^{19}^{19}19In fact, one can take , , , and define the measure by . Notice also that mean convergence for the averages follows if we prove mean convergence for all averages . Combining all these observations, we can easily reduce matters to proving mean convergence for the averages when all functions are measurable. This is a significant simplification of our original problem, and in fact it is now straightforward to deduce the required convergence property from the mean ergodic theorem.
This approach has proved particularly useful for handling convergence problems of multiple ergodic averages of commuting transformations with linear iterates that previously seemed intractable [9, 111, 10, 59, 60] (see also [13] for an application to continuous time averages). A drawback is that it does not give much information about the limiting function, and also, up to now, it has not proved to be as useful when some of the iterates are nonlinear (for polynomial iterates though there is some progress in this direction [11, 12]).
2.8. Furstenberg correspondence principle
We frequently use the following correspondence principle of Furstenberg [91, 92] (the formulation given is from [23]) in order to reformulate statements in combinatorics as multiple recurrence statements in ergodic theory:
Furstenberg Correspondence Principle.
Let , be a set of integers, and . Then there exist a probability space , commuting invertible measure preserving transformations , and a set , with , and such that
(8) 
for every . Furthermore, if one can take .
We are going to mention several applications of this principle in the next two subsections.
2.9. Equivalent problems for sequences
It turns out, and sometimes it is useful to be aware of this observation, that problems about mean convergence and multiple recurrence in ergodic theory are intimately related with similar problems involving bounded sequences of complex numbers. We give some explicit examples below.
Given a collection of sequences of integers , it turns out that the following two properties are equivalent:

For every invertible measure preserving system , and with , there exists , such that

For every nonnegative bounded sequence with , there exists , such that
Using the correspondence principle of Furstenberg it is not hard to see that the first statement implies the second. To see that the second statement implies the first it suffices to set for a suitable point (almost every works) and use the mean ergodic theorem.
For convergence problems it is convenient to define the following notion: a sequence of complex numbers is stationary with respect to the sequence of intervals , if for every and , the averages
converge as . Using a diagonal argument it is easy to show that any bounded sequence of complex numbers is stationary with respect to some subsequence of intervals. Given a collection of sequences of integers , it turns out that the following two properties are equivalent:

For every invertible measure preserving system , and nonnegative , the averages
converge as .

For every bounded sequence of complex numbers, stationary with respect to the sequence of intervals , the averages
converge as .
One can get similar statements for mean convergence, as well as for convergence and recurrence properties of commuting transformations (in this case one has to use sequences in several variables).
3. Some useful notions
To ease our exposition we collect some notions that are frequently used in subsequent sections.
3.1. Recurrence
The next notions are used to describe multiple recurrence properties of a single sequence:
Definition 3.1.
We say that the sequence of integers is

good for recurrence of powers if for every , every invertible measure preserving system , and set with , we have
for some with .^{20}^{20}20We remark, that for this and subsequent statements, the existence of a single for which the multiple intersection has positive measure, forces the existence of infinitely many with the same property (but not necessarily for a set of with positive upper density, although more often than not this is also true).

good for multiple recurrence of powers if it is good for recurrence of powers for every .

good for recurrence of commuting transformations if for every probability space , commuting invertible measure preserving transformations , and set with , we have
for some with .

good for multiple recurrence of commuting transformations if it is good for recurrence of commuting transformations for every .
The fact that the sequence is good for multiple recurrence of powers corresponds to the multiple recurrence result of H. Furstenberg [91], and the fact that it is good for multiple recurrence of commuting transformations corresponds to the multidimensional extension of this result of H. Furstenberg and Y. Katznelson [97]. Further examples of sequences that are good for multiple recurrence of commuting transformations include: integer polynomials with zero constant term [35], the shifted primes ([191] for powers and [42] in general), integer polynomials with zero constant term evaluated at the shifted primes ([191] for powers and [81] in general), several generalized polynomial sequences [44], and some random sequences of zero density [88]. The sequence , where is positive, is known to be good for multiple recurrence of powers [90] (see also [79]), but if is greater than , it is not known whether it is good for multiple recurrence of commuting transformations. Examples of sequences that are good for recurrence of powers but are not good for recurrence of powers can be found in [86].
Let us also mention at this point some obstructions to recurrence. One can check that if the sequence is good for recurrence, then the equation has a solution for every ; as a consequence, the sequences , , , are not good for recurrence. More generally, if the sequence is good for recurrence, then for every and the inequality has a solution (letting gives the previous congruence condition); as a consequence, the sequence is not good for recurrence (try and ). This obstruction also implies that if a sequence is lacunary, meaning, , then it is not good for recurrence, since it is well known that in this case there exist an irrational and a positive such that for every .
Furthermore, if a sequence is good for recurrence of powers, then for every and the inequality has a solution (this is implicit in [92] Section 9.1, and is proved in detail in [86]). It follows that if one forms a sequence by putting the elements of the set in increasing order, then this sequence is not going to be good for recurrence of powers (one can show that it is going to be good for recurrence [86]). And there are also other more restrictive obstructions, if the sequence is good for recurrence of powers, then for every and the inequality has a solution. More generally, for recurrence of powers, one can state further obstructions by using higher degree polynomials with zero constant term, or more complicated generalized polynomials.
The next notions are used to describe multiple recurrence properties of a collection of sequences:
Definition 3.2.
We say that the collection of sequences of integers is

good for recurrence of a single transformation if for every invertible measure preserving system , and set with , we have
for some with for .

good for recurrence of commuting transformations if for every probability space , commuting invertible measure preserving transformations , and set with , we have
for some with for .
Examples of collections of sequences that are known to be good for recurrence of commuting transformations include collections of: integer polynomials with zero constant term [35] (for arbitrary integer polynomials necessary and sufficient conditions where given in [40]) and integer polynomials with zero constant term evaluated at the shifted primes [81]. For every , the collection , where , is known to be good for recurrence of a single transformation [79], but it is not known whether it is always good for recurrence of commuting transformations.
The correspondence principle of Furstenberg enables us to deduce statements in density Ramsey theory from multiple recurrence statements in ergodic theory. Using this principle and some elementary arguments one is able to reformulate the previous ergodic notions to purely combinatorial ones:
Theorem.
The sequence of integers is

good for recurrence of powers if and only if for every , every set with contains patterns of the form
for some and with .

good for recurrence of commuting transformations if and only if for every and , every set with contains patterns of the form
for some and with .
For instance, the fact that the sequence is good for multiple recurrence of powers corresponds to the theorem of Szemerédi on arithmetic progressions.
Theorem.
The collection of sequences of integers is

good for recurrence of a single transformation if and only if every set with contains patterns of the form
for some and with for .

good for recurrence of commuting transformations if and only if for every , every set with contains patterns of the form
for some and with for .
Let us also remark that the previous notions admit equivalent uniform versions that are often useful for applications. For instance, one can prove the following (see [34] for an argument that works for polynomials and [87] for an argument that works for general sequences):
Theorem.
The collection of sequences of integers is good for recurrence of a single transformation if and only if
For every there exist and , such that for every and integer set with , we have
for some .
For every there exist and , such that for every invertible measure preserving system and with , we have that
for some .
3.2. Convergence
The next notion is used to describe multiple convergence properties of a single sequence:
Definition 3.3.
We say that the sequence of integers is

good for convergence of powers (of a single transformation) if for every , invertible measure preserving system , and functions , the averages
converge in the mean.

good for multiple convergence of powers (of a single transformation) if it is good for convergence of powers for every .

good for convergence of commuting transformations if for every probability space , commuting invertible measure preserving transformations , and functions , the averages
converge in the mean.

good for multiple convergence of commuting transformations if it is good for convergence of commuting transformations for every .
Examples of sequences that are good for multiple convergence of commuting transformations include: the sequence [181] (see also [186, 9, 111]), the sequence of primes [81], and some random sequences of zero density [88]. Additional examples of sequences that are known to be good for multiple convergence of powers include: sequences given by integer polynomials [116, 141], integer polynomials evaluated at the primes [191] (see also [81]), and sequences of the form