Cubic averages and large intersections

We discuss the nature of the phenomenon of “large limits along cubic averages” and obtain some rather general new results. We also prove a general version of Furstenberg’s correspondence principle that applies to uncountable amenable discrete groups and utilize this to derive combinatorial corollaries of ergodic results involving large limits. 0. Introduction Let (X,B, μ, T ) be an invertible probability measure preserving system. The classical Poincaré recurrence theorem (see [Po], Theorem I) states that for any A ∈ B with μ(A) > 0 there exists n ∈ N such that μ(A ∩ TA) > 0. For mixing systems one has limn→∞ μ(A ∩ TA) = μ(A), and it is natural to ask whether for any (X,B, μ, T ), any A ∈ B, and any δ > 0 one can find n 6= 0 such that μ(A ∩ TA) ≥ μ(A) − δ, and, if so, how “large” the set Rδ(A) = { n ∈ Z : μ(A ∩ TA) ≥ μ(A) − δ } can be. These questions were addressed in [Kh], where the following result, often quoted as “Khintchine’s recurrence theorem”, was obtained. Theorem 0.1. For any invertible probability measure preserving system (X,B, μ, T ), any A ∈ B, and any δ > 0, the set Rδ(A) is syndetic. (1) (We are taking the liberty to formulate Khintchine’s result for powers of a single measure preserving transformation. Khintchine’s paper dealt with continuous measure preserving flows.) In [Kh], Theorem 0.1 was derived from the following stronger result. (Again, we formulate it for Zrather than for R-actions.) Partially supported by NSF grants DMS-0901106 and DMS-1162073. (1) A set S in a topological group G is said to be (left) syndetic if there exists a compact set F ⊆ G such that G = FS. If G is a discrete group, F has to be finite. 1 Theorem 0.2. For any invertible probability measure preserving system (X,B, μ, T ) and any A ∈ B, one has lim N−M→∞ 1 N −M N−1 ∑ n=M μ(A ∩ TA) ≥ μ(A). By iterating Khintchine’s theorem, one immediately observes that for any n ∈ Rδ(A) there exists a syndetic set of m ∈ Z such that μ ( (A ∩ TA) ∩ T(A ∩ TA) ) = μ(A ∩ TA ∩ TA ∩ TA) > μ(A) − 2δ. (0.1) This leads to the natural question of whether, for any δ > 0, the set of pairs (n,m) satisfying (0.1) is syndetic in Z. The affirmative answer follows from the following theorem. Theorem 0.3. ([B1]) For any invertible probability measure preserving system (X,B, μ, T ) and any A ∈ B, one has lim N1−M1,N2−M2→∞ 1 (N1 −M1)(N2 −M2) ∑ M1≤n<N1 M2≤m<N2 μ(A ∩ TA ∩ TA ∩ TA) ≥ μ(A). Corollary 0.4. For any invertible probability measure preserving system (X,B, μ, T ), any A ∈ B, and any δ > 0, the set { (n,m) ∈ Z : μ(A ∩ TA ∩ TA ∩ TA) > μ(A) − δ } is syndetic. Theorem 0.3 was extended in [HK1] and [HK2] to multiparameter expressions of the form μ (⋂ ε1,.,εk∈{0,1} T 11kkA ) , n = (n1, . . . , nk) ∈ Z , which in turn implies, for any δ > 0, the syndeticity of the set { n ∈ Z : μ ( ⋂ ε1,.,εk∈{0,1} T 11kkA ) > μ(A) k − δ } . The proofs of these results in [HK1] and [HK2] utilize in a rather crucial way knowledge about characteristic factors responsible for the limiting behavior of the expressions 1 (N1−M1)···(Nk−Mk) ∑ M1≤n1<N1 . Mk≤nk<Nk ∏ ε1,. ,εk∈{0,1} T 11kkf , and it is of interest to attempt to establish the results on large intersections in the situations where there is (so far) no sufficient knowledge on the structure of the corresponding characteristic factors. In particular, one would like to know if, given commuting invertible measure preserving (2) A factor Y of X is characteristic if the limit, as Ni −Mi → ∞, of the expressions under consideration depends not on f , but only on E(f |Y ) – the conditional expectation of f with respect to Y (the orthogonal projection of f on L(Y ) ⊆ L(X)). 2 transformations T1, . , Tk of a probability space (X,B, μ) and a set A ∈ B with μ(A) > 0, the set R (k) δ (A) = { (n1, . . . , nk) ∈ Z k : μ ( ⋂ ε1,.,εk∈{0,1} T ε1n1 1 . . . T εknk k A ) > μ(A) k − δ } (0.2) is syndetic. The goal of this short paper is to show that the phenomenon of large limits along cubic configurations has a very general scope and hinges on a simple “Fubini principle” (Lemma 1.1 below), which, roughly, states that the limits of uniform multiparameter Cesàro averages can be replaced by iterated limits of uniform Cesàro averages. In what follows we will establish a general scheme that can be viewed as a machine for obtaining various results on large limits of cubic averages. This machine has an input and an output. The input consists of two convergence statements: (i) a general multiparameter limiting theorem that stipulates the existence of certain uniform Cesàro limits (but does not require any description of these limits); (ii) an ergodic limiting result pertaining to single or multiple recurrence that guarantees a “large” uniform Cesàro limit. An example of such a result is given by Theorem 0.2. (We will provide below a few more examples of various degrees of generality.) Then the machine’s output is a theorem on large limits of cubic averages. An example of such a theorem is Theorem 0.3, but, as we will see, significantly more general theorems hold true as well. To help the reader to get a feeling about the general results which we obtain in this paper, let us give a sketch of the proof of the following fact, which is a rather special case of Theorem 0.8 below. Theorem 0.5. Let T1, . , Tk be invertible transformations of a probability measure space (X,B, μ), which commute (or, more generally, generate a nilpotent group). Then for any A ∈ B, lim N1−M1,. ,Nk−Mk→∞ 1 (N1 −M1) · · · (Nk −Mk) ∑ M1≤n1<N1 . Mk≤nk<Nk μ ( ⋂ ε1,.,εk∈{0,1} T ε1n1 1 · · ·T εknk k A ) ≥ μ(A) k , and hence the set R (k) δ (A) defined in (0.2) is syndetic. Proof sketch. The input statement of the form (i) that is needed for the proof is the following fact (see [W], Theorem 5.1): • For any A ∈ B, the limit lim N1−M1,. ,Nk−Mk→∞ 1 (N1 −M1) · · · (Nk −Mk) ∑ M1≤n1<N1 . Mk≤nk<Nk μ ( ⋂ ε1,.,εk∈{0,1} T ε1n1 1 · · ·T εknk k A )