A spectral refinement of the Bergelson-Host-Kra decomposition and new multiple ergodic theorems
We investigate how spectral properties of a measure preserving system are reflected in the multiple ergodic averages arising from that system. For certain sequences we provide natural conditions on the spectrum such that for all ,
in -norm. In particular, our results apply to infinite arithmetic progressions , Beatty sequences , the sequence of squarefree numbers , and the sequence of prime numbers .
We also obtain a new refinement of Szemerédi’s theorem via Furstenberg’s correspondence principle.
Let be a probability space and let be a measure preserving transformation. The discrete spectrum of the measure preserving system is the set of eigenvalues for which there exists a non-zero eigenfunction satisfying , where . It follows from the spectral theorem that given any functions , there exists a complex measure on the torus such that
Decomposing the measure into its discrete and continuous components , one can then represent the single-correlation sequence as
where is a Bohr almost periodic sequence111A sequence is Bohr almost periodic if there exists a compact abelian group (written multiplicatively), elements and a continuous function such that . and is a null-sequence222A sequence is a null-sequence if .. It is a further consequence of the spectral theorem that for any frequency which does not belong to the discrete spectrum , one has and hence
This observation can be reformulated as
where denotes the spectrum of the sequence in the sense introduced by Rauzy in , which we recall now.
The spectrum of an arbitrary bounded sequence is the set of frequencies for which
Throughout this paper we are only concerned with sequences for which the limit supremum in the above expression is an actual limit.
Formula (1.2) serves as the premise of our paper. Its importance lies in the many variations of the mean ergodic theorem that one can derive from it. For instance, for any , if the discrete spectrum of an ergodic system is disjoint from then it follows from (1.2) that for any and any ,
More generally, for any real number , if the discrete spectrum of an ergodic system is disjoint from then for any and any ,
where denotes the prime-counting function.
In this paper we seek to extend (1.2) from single-correlation sequences to multi-correlation sequences, i.e., sequences of the form
The theory of multi-correlation sequences was pioneered by Furstenberg in connection with his ergodic-theoretic proof of Szemerédi’s theorem . A result of Bergelson, Host and Kra  offers a decomposition of multi-correlation sequences in analogy with (1.1):
where is a null-sequence and is a nilsequence333A sequence is a nilsequence if it can be approximated in by sequences of the form , where is a continuous function on the compact homogenous space of a nilpotent Lie group , and .. By examining the spectrum of the nilsystem from which the nilsequence in (1.6) arises, we show that the spectrum of the multicorrelation sequence is contained in the discrete spectrum of its originating system (cf. creftype 2.1 below). From this we derive several multiple ergodic theorems (see Theorems 2.5, 2.7, 2.9 and 2.10 in Section 2). As corollaries of Theorems 2.5 and 2.9, we obtain the following generalizations of (1.4) and (1.5): For any real number , if the discrete spectrum of an ergodic system is disjoint from then for any , and any
For any totally ergodic system , any and any ,
Structure of the paper:
In Section 2 we formulate the main results of this paper and present relevant examples as well as applications to combinatorics. In Section 3 we provide the necessary background on the theory of nilmanifolds and nilsystems, which is used in the rest of the paper.
Our main technical result, creftype 2.1, is proven in three steps: in Section 5, we reduce it to the special case of nilsystems. In Section 4 we derive a proof of this special case conditionally on a result involving the spectrum of nilsystems. The proof of the latter is provided in Section 7. In the remainder of Section 5 and also in Section 6 we deduce the other results stated in Section 2 from creftype 2.1.
We thank V. Bergelson for many inspiring conversations, N. Frantzikinakis, B. Kra and A. Leibman for providing helpful references, and D. Glasscock for helpful comments regarding an earlier draft of this paper. The authors also thank the referee for several pertinent suggestions which greatly improved the final version of this paper.
2 Statement of results
In this section we state the main results of the paper; the proofs are presented in Sections 5 and 6. The following theorem is our main technical result. For a definition of nilsystems, see Section 3.
Let , let be an ergodic measure preserving system and let . For every there exists a decomposition of the form
where is a null-sequence, satisfies and for some and , where is a -step nilsystem whose discrete spectrum is contained in the discrete spectrum of .
A natural question is whether analogues of creftype 2.1 hold for commuting transformations or for polynomial iterates. However these extensions seem to be out of reach by the methods used in the current paper.
From creftype 2.1 we derive various multiple ergodic theorems. The first theorem we derive this way is an extension of (1.3). An equivalent result was proven by Frantzikinakis in . In the following we will use to denote the subgroup of generated by a subset . Subsets of are tacitly identified with their projections onto .
Theorem 2.4 (cf. [8, Theorem 6.4]).
Let , and let be an ergodic measure preserving system whose discrete spectrum satisfies . For any ,
where convergence takes place in . In particular, if is totally ergodic, then equation (2.1) holds for all .
creftype 2.4 features multi-correlation sequences along infinite arithmetic progressions . The next theorem is a generalization in which infinite arithmetic progressions are replaced by more general Beatty sequences .
Let with , and let be an ergodic measure preserving system whose discrete spectrum satisfies . For any ,
where convergence takes place in . In particular, since discrete spectra are always countable, we have that for any fixed system and for almost all equation (2.2) holds for all .
creftype 2.5, together with a standard application of Furstenberg’s correspondence principle (cf. [3, Proposition 3.1]), implies the following combinatorial result. Recall that the upper density of a set is defined by
Let , and let have positive upper density. Then for almost every and every there exists a -term arithmetic progression in with common difference in the Beatty sequence .
In fact, under the assumptions of creftype 2.6, there are many arithmetic progressions contained in with common difference in a Beatty sequence. More precisely, there is a syndetic444A subset is called syndetic if it has bounded gaps, more precisely if there exists such that any interval of length contains at least one element of . set such that for every , there exist a set with positive upper density and with the property that for every , the set is contained in .
Recall that a bounded sequence is Besicovitch almost periodic if for every , there exists a trigonometric polynomial , where , and , such that
The indicator function of a Beatty sequence is a Besicovitch almost periodic sequence with spectrum contained in the subgroup . Thus, sacrificing the uniformity in the Cesàro averages on the left hand side of (2.1) and (2.2), one can extend Theorems 2.4 and 2.5 as follows.
Let be a Besicovitch almost periodic sequence, and let be a measure preserving system whose discrete spectrum satisfies . For any ,
creftype 2.7 is not true for uniform Cesàro averages. One way of obtaining a version of creftype 2.7 with uniform Cesàro averages is by replacing Besicovitch almost periodic sequences with Weyl almost periodic sequences555A bounded sequence is Weyl almost periodic if for every , there exists a trigonometric polynomial such that . In fact, one can easily modify the proof of creftype 2.7 given below to obtain a proof of this variation.
An interesting application of creftype 2.7 concerns the sequence of squarefree numbers. Since the indicator function of the set of squarefree numbers is Besicovitch almost periodic with rational spectrum (cf. Section 3.4), it follows that for any totally ergodic ,
By combining creftype 2.1 with results of Green, Tao and Ziegler [17, 18, 19] on the asymptotic Gowers uniformity of the von Mangoldt function, we obtain the following multiple ergodic theorem along primes for totally ergodic systems.
Let and let be a totally ergodic system. For every ,
The case of creftype 2.9 was obtained by Frantzikinakis, Host and Kra in [11, Theorem 5]. In the same paper they outline the proof of creftype 2.9 in full generality, conditional on the then unknown creftype 6.3.
Using creftype 2.7, we obtain a strengthening of the above result involving primes in Beatty sequences. Let and let .
Let with irrational and let be a measure preserving system whose discrete spectrum satisfies . For every ,
In particular, if is totally ergodic then for almost all equation (2.4) holds for all .
In this section we give an overview of the theory of nilspaces and nilmanifolds.
3.1 Nilmanifolds and sub-nilmanifolds
Let be a Lie group with identity . The lower central series of is the sequence
where is, as usual, the subgroup of generated by all the commutators with and . If for some finite we say that is (-step) nilpotent. Each is a closed normal subgroup of (cf. [27, Section 2.11]).
Given a nilpotent Lie group and a uniform666A closed subgroup of is called uniform if is compact or, equivalently, if there exists a compact set such that . and discrete subgroup of , the quotient space is called a nilmanifold. Naturally, acts continuously and transitively on via left-multiplication.
Any element with the property that for some is called rational (or rational with respect to ). A closed subgroup of is then called rational (or rational with respect to ) if rational elements are dense in . For example, the subgroups in the lower central series of are rational with respect to any uniform and discrete subgroup of . (A proof of this fact can be found in [34, Corollary 1 of Theorem 2.1] for connected and in [27, Section 2.11] for the general case.)
It is shown in  that a closed subgroup is rational if and only if is a uniform discrete subgroup of if and only if is closed in .
If is a nilmanifold, then a sub-nilmanifold of is any closed set of the form , where and where is a closed subgroup of . It is not true that for every closed subgroup of and for every element in the set is a sub-nilmanifold of ; as a matter of fact, from creftype 3.1 it follows that is closed in (and hence a sub-nilmanifold) if and only if the subgroup is rational with respect to .
3.2 Nilsystems and their dynamics
Let be a -step nilpotent Lie group and let be a nilmanifold. In the following we will use or ( if we want to emphasize the dependence on ) to denote the translation by a fixed element , i.e. . The map is called a nilrotation and the pair is called a (-step) nilsystem.
Every nilmanifold possesses a unique -invariant probability measure called the Haar measure on X (cf. [34, Lemma 1.4]). We will use to denote this measure.
Let us state some classical results regarding the dynamics of nilrotations.
Suppose is a nilsystem. Then the following are equivalent:
is transitive777A topological dynamical system is called transitive if there exists at least one point with dense orbit.;
is strictly ergodic888A topological dynamical system is called strictly ergodic if there exists a unique -invariant probability measure on and additionally the orbit of every point in is dense.;
Moreover, the following are equivalent:
is connected and is ergodic.
is totally ergodic.
A theorem by Lesigne  asserts that for any nilmanifold with connected and any the closure of the set is a sub-nilmanifold of . (Actually, he shows that the sequence equidistributes with respect to the Haar measure on some sub-nilmanifold of , but in virtue of creftype 3.2 these two assertions are equivalent.) Leibman has extended this result as follows.
Theorem 3.3 ([26, Corollary 1.9]).
Let be a nilpotent Lie group and let be a uniform and discrete subgroup. Assume is a connected sub-nilmanifold of and . Then is a disjoint union of finitely many connected sub-nilmanifolds of .
3.3 The Kronecker factor of a nilsystem
Let be a nilmanifold and let be a normal, closed and rational subgroup of . Since is closed, the quotient topology on is Hausdorff and the map that sends elements to their right cosets is continuous and commutes with the action of . Therefore the nilsystem is a factor of with factor map .
An important tool in studying equidistribution of orbits on nilmanifolds is a theorem by Leon Green (see [1, 20, 33]). In  Leibman offers a refinement of this classical result of Green, a special case of which we state now. Here and throughout the text we denote by the connected component of containing the group identity .
Theorem 3.4 (cf. [27, Theorem 2.17]).
Let be a connected nilmanifold, let and let , where denotes the group generated by and . Then is ergodic on if and only if is ergodic on .
Note that in creftype 3.4 it is not explicitly stated but implied that is a normal, closed and rational subgroup of and hence the factor space is well defined.
Given a measure preserving dynamical system let denote the smallest sub--algebra of such that any eigenfunction of becomes measurable with respect to . The resulting factor system is called the (measure-theoretic) Kronecker factor of .
The following corollary of creftype 3.4 describes the Kronecker factor of a connected ergodic nilsystem.
Let be a connected nilmanifold, let and assume is ergodic. Define . Then the Kronecker factor of is .
For the proof of creftype 3.5 it will be convenient to recall the definition of vertical characters: Let be a connected nilmanifold and let be the lower central series of . The quotient is a connected compact abelian group and hence isomorphic to a torus . We call the vertical torus of . Since is contained in the center of , the vertical torus acts naturally on . A measurable function is called a vertical character if there exists a continuous group character of such that for all and almost every .
Proof of creftype 3.5.
Notice that the nilsystem is isomorphic to the nilsystem , where . We can therefore assume without loss of generality that . We proceed by induction on the nilpotency class of . Suppose is a -step nilpotent Lie group. If , is abelian and the result is trivial. Next, assume that and that creftype 3.5 has already been proven for all nilpotent Lie groups of step .
Observe that is a compact group and hence is contained in the Kronecker factor of . It thus suffices to show that for all eigenfunctions of the system one has
Let be an eigenvalue of the Koopman operator associated with , let be its (non-trivial) eigenspace and let .
Let denote the vertical torus of and note that the action of on commutes with the action of . In particular, leaves the eigenspace invariant. It thus follows from the Peter-Weyl theorem that decomposes into a direct sum of eigenspaces for the Koopman representation of . In other words, any -eigenfunction can be further decomposed into a sum of vertical characters that are also contained in . It therefore suffices to establish (3.1) in the special case where is a vertical character.
Now assume , is a group character of and for all and almost every . We distinguish two cases; the first case where is trivial and the second case where is non-trivial.
Let us first assume that is trivial, i.e. for all . This implies that is invariant. Let denote the nilpotent Lie group and let denote the natural quotient map. We define , which is a uniform and discrete subgroup of , and we define . Since is invariant it can be identified with a function on the nilmanifold and is then an eigenfunction for , where . Since is an -step nilpotent Lie group, we can invoke the induction hypothesis and conclude that
Now assume that is non-trivial. Since is connected, any non-trivial character has full range in the unit circle. In particular, there exists such that . Pick any element such that and define . Then from and from it follows that . Also, note that since the actions of and on the factor are identical (because ), it follows from the ergodicity of that acts ergodically on . Finally, the groups and are identical and hence it follows from creftype 3.4 that the ergodicity of lifts from to . We conclude that has to be a constant function, thereby satisfying (3.1). ∎
3.4 Spectrum of almost periodic sequences
In this section we collect a few facts about the spectrum of almost periodic sequences; we refer the reader to the book of Besicovitch  for a complete treatment on the theory of almost periodic sequences.
Almost periodic sequences were first introduced by Bohr in . In his second paper on this subject  he proves that any Bohr almost periodic sequence can be approximated uniformly by trigonometric polynomials whose frequencies are all contained in the spectrum . This theorem is known as Bohr’s approximation theorem. An analogue of Bohr’s approximation theorem for Besicovitch almost periodic sequences was later obtained by Besicovitch. He showed that the spectrum of a Besicovitch almost periodic sequence is at most countable and then proved that any Besicovitch almost periodic sequence can be approximated in the Besicovitch seminorm by trigonometric polynomials whose frequencies are all contained in the spectrum . The precise statement of Besicovitch’s result is as follows999In his book, Besicovitch only deals with continuous almost periodic functions (i.e., almost periodic functions with domain ), but the proof of [4, Theorem II.8.2Â] also works for discrete almost periodic sequences (i.e., almost periodic functions with domain ); see also [2, Section 3]..
Theorem 3.6 (cf. [4, Theorem II.8.2Â(page 105)]).
Let be a Besicovitch almost periodic sequence with spectrum . Then for every there exists a trigonometric polynomial with and such that
We will also make use of the following lemma regarding the spectrum of the product of two Besicovitch almost periodic sequences.
Let be bounded Besicovitch almost periodic sequences. The the product is also Besicovitch almost periodic with spectrum .
In view of creftype 3.6, we can approximate each with a trigonometric polynomial whose spectrum is contained in . Observe that the product is a trigonometric polynomial with spectrum contained in . Finally, it is not hard to show that approximates , which finishes the proof. ∎
4 Proving creftype 2.1 for the special case of nilsystems
Let , let be an ergodic -step nilsystem and let . Then
where is a null-sequence and for some and , where is a -step nilsystem whose discrete spectrum is contained in , the discrete spectrum of .
The main new ingredient used in the proof of creftype 4.1 is the following result.
Let , let be a connected nilmanifold and let be an ergodic nilrotation. Define and
Let be a nilmanifold and let denote the natural projection of onto . We will use to denote the point . Consider a closed subgroup of . As noticed in creftype 3.1 the set is a sub-nilmanifold of if and only if is rational. Let denote the normal closure of in , that is, let be the smallest normal subgroup of containing . One can show that if is closed and rational then so is (cf. ). In particular, the set is a sub-nilmanifold of containing . We call the normal closure of .
Note, every sub-nilmanifold of can be viewed as a nilmanifold on its own and in particular it has its own Haar measure . Moreover, for any , the Haar measure of the sub-nilmanifold coincides with the push forward of under .
Proposition 4.3 (cf. [29, Proposition 3.1]).
Assume is a connected nilmanifold, is the natural projection of onto and is a connected sub-nilmanifold of containing . Let and assume is dense in . If denotes the normal closure of , then for all we have
Let be a nilsystem, let be a connected sub-nilmanifold containing the origin and assume that is also a connected sub-nilmanifold of . Then there exists a factor of , and a point such that for any continuous function , there exists such that
Since is invariant under and , we can find a closed rational subgroup of such that . Therefore, is a uniform discrete subgroup of and the nilsystem is isomorphic to . In the following we will identify with and vice versa. Let be the normal closure of the sub-nilmanifold in and let denote the corresponding normal subgroup of such that .
Define and let denote the natural projection. As explained at the beginning of Section 3.3, is a well defined factor of with factor map .
To prove creftype 4.1 we will also require a technical lemma:
Let be an ergodic connected nilsystem of step and let . Then there exists an ergodic nilsystem of step with exactly connected components and such that the restriction of to each connected component of yields a system isomorphic to .
First, we claim that one can embed the connected nilsystem into a nilflow , so that is a subnilmanifold of invariant under . Indeed, say . One can assume that the identity component of is simply connected, by passing to the universal cover if needed. Next one can use [34, Theorem 2.20] to find a connected simply connected nilpotent Lie group such that and is a uniform discrete subgroup of . In particular is a sub-nilmanifold of . Since is connected and simply connected, for any element the associated one-parameter subgroup is well defined (cf. [27, Subsection 2.4]). In particular, the nilrotation can be extended to a nilflow on by defining for all , .
Next, consider the product nilsystem , so that as a nilmanifold and the nilrotation is defied as . Finally, let be the orbit of under . Since preserves , we deduce that has precisely connected components. In fact,