New York Journal of Mathematics

New York J. Math. 3A (1998) 89–98.

Weighted Ergodic Theorems Along Subsequences of Density Zero

Roger L. Jones, Michael Lin, and James Olsen

Abstract. We consider subsequence versions of weighted ergodic theorems, and show that for a wide class of subsequences along which a.e. convergence of Cesaro averages has been established, we also have a.e. convergence for the subsequence Cesaro weighted averages, when the weights are obtained from uniform sequences produced by a connected apparatus.

Contents

1. Introduction 89 2. Besicovitch Weights 90 3. Uniform Sequences 93 References 97

1. Introduction

Let (X, F, μ) be a probability space. For T a linear contraction of L_{p}(X, F, μ) = L_{p}, p ≥ 1, various ergodic theorems consider the a.e. convergence of the averages \frac{1}{N} ∑_{k=1}^{N} T^{k}f(x) for every f ∈ L_{p}. More generally, for \{n_{k}\} an increasing sequence of positive integers, various authors have considered the a.e. convergence of averages of the form \frac{1}{N} ∑_{k=1}^{N} T^{n_{k}}f(x). When \{n_{k}\} has positive density, this convergence can be represented (e.g., [3]) as convergence of weighted averages \frac{1}{N} ∑_{k=1}^{N} a(k)T^{k}f(x), with {a(k)} a 0-1 sequence. We will be interested in subsequence versions of these weighted averages. That is, for a sequence {a(k)} of complex numbers for which the weighted averages converge, we will be interested in studying the almost everywhere convergence of subsequence averages of the form

\frac{1}{N} ∑_{k=1}^{N} a(n_{k})T^{n_{k}}f. (1)

\begin{matrix}c \\ ◯ \end{matrix}1998 State University of New York ISSN 1076-9803/98

Received November 21, 1997.

Mathematics Subject Classification. 47A35; 28A65.

Key words and phrases. Besicovitch sequences, uniform sequences, Dunford-Schwartz operators, amplitude modulation, pointwise subsequence ergodic theorem.

R. Jones is partially supported by NSF Grant DMS—9531526.

M. Lin is partially supported by the Israel Science Foundation.

J. Olsen is partially supported by ND EPSCoR through NSF Grant # OSR-5452892.

The types of integer sequences {n_{k}} which we will be interested in are those for which we have a.e. convergence of the unweighted Cesaro averages in the measure preserving case, that is, for fixed p > 1, sequences for which the averages \frac{1}{N} ∑_{k=1}^{N} f(τ^{n_{k}}x) converge a.e. for all f in L_{p}, and for all measure preserving transformations τ. Such sequences include the sequence {n_{k}} where n_{k} = k^{2}, or, more generally, n_{k} = k^{t}, t a positive integer, or n_{k} = k-th prime, or any of the sequences studied in [5] or [20], as well as a variety of other sequences. We will call such sequences good universal in L_{p}. If {n_{k}} is good universal in L_{p}, and also has the property that for every measure preserving transformation τ on a non-atomic probability space, the maximal operator f^{⋆}(x) = sup_{N} \frac{1}{N} ∑_{k=1}^{N} |f(τ^{n_{k}}x)| is strong type (p, p) (that is, ∥f^{⋆}∥_{p} ≤ c_{p}∥f∥_{p} for every f ∈ L_{p}), then we will say that the sequence {n_{k}} is strongly good universal in L_{p}.

The types of operators we will consider are those induced by measure preserving point transformations (i.e., T f = f ∘ τ, where τ is a measure preserving point transformation of X), Dunford-Schwartz operators (i.e., linear operators of L_{p}, all p, 1 ≤ p ≤ ∞ such that ∥T ∥_{∞} ≤ 1 and ∥T ∥_{1} ≤ 1), and positively dominated contractions of L_{p}, p fixed, 1 < p < ∞ (i.e., an operator T of L_{p} such that there is a positive operator S of L_{p} norm less than or equal to one that takes non-negative functions to non-negative functions and |T f(x)| ≤ S|f|(x) a.e.).

When the limit of the averages given in (1) exists for all f ∈ L_{p} for a particular sequence {n_{k}}, a particular sequence of weights {a(k)}, and all T in some class C of operators of L_{p}, we will say that {a(k)} is a good weight sequence along {n_{k}} for C on L_{p}. In this terminology, we know that the sequence {a(k) = 1} is a good weight sequence along {n_{k}} for C when {n_{k}} is good universal in L_{p}, p > 1, and C is the class of measure preserving transformations. Moreover, for the sequences {n_{k}} mentioned earlier and p > 1, we can enlarge the class C to include the operators mentioned above ([4], [5], [20], [11], [13]). We will investigate how much of this is true for some other previously considered sequences of weights {a(k)}, which are good weights along {n_{k} = k}.

2. Besicovitch Weights

For a sequence of complex numbers {a(k)}, define for 1 ≤ p < ∞, the p seminorms

∥\{a(k)\}∥_{p} = (lim sup_{N→∞} \frac{1}{N} ∑_{k=1}^{N} |a(k)|^{p})^{\frac{1}{p}} .

If {a(k)} is defined by a(k) = ∑_{j=1}^{m} b_{j}λ_{j}^{k}, where λ_{j}, j = 1, …, m are complex numbers of modulus one and b_{j} are complex numbers , we call {a(k)} a trigonometric polynomial. The p-Besicovitch sequences will be the closure in the p-semi-norm of the trigonometric polynomials. Besicovitch sequences, as good sequences of weights, have already been extensively studied. We give just a few of the references that contain some of the results we will need ([17], [12], [15], [19], [3]).

Note that the p-semi-norm of a bounded sequence does not change if the values of the sequence are changed, in a bounded way, on a subsequence of the integers of density zero. It is clear that a set of trigonometric polynomials can be used to approximate bounded functions that exhibit any behavior whatever along a sequence of density zero. Therefore, we cannot in general expect a Besicovitch sequence {a(k)} to be a good weight sequence along good universal sequences {n_{k}} of density zero.

In [15], Besicovitch sequences defined only on subsets of the integers are introduced. In the terminology introduced there, the class B_{p,{n_{k}}} of p, {n_{k}}-Besicovitch sequences is defined to be the closure of the trigonometric polynomials in the p semi-norm defined by

∥\{a(k)\}∥_{p,\{n_{k}\}}^{p} = lim sup_{N→∞} \frac{1}{N} ∑_{k=1}^{N} |a(n_{k})|^{p} .

Since the integers are an abelian group, the closure of the trigonometric polynomials is the same as the closure of the almost periodic functions on the integers (see [15]).

Unfortunately, when we consider good universal sequences of zero density, the measures on the integers induced by such sequences {n_{k}}, i.e., the measures μ_{N} that give the measure \frac{1}{N} to the first N terms of the sequence \{n_{k}\} and zero to the rest of the integers, are not ergodic. Hence, most of the results of [15] will not apply. We do have, however, that for a fixed {n_{k}}, all the B_{p,{n_{k}}} classes contain the same bounded sequences, that is, B_{p,{n_{k}}} ∩ℓ_{∞} = B_{1,{n_{k}}} ∩ ℓ_{∞} for all p, 1 ≤ p < ∞ ([15], Theorem 2.1). We will refer to this class as bounded {n_{k}}-Besicovitch sequences.

Routine arguments give the following results.

Theorem 2.1. Fix p, 1 ≤ p < ∞. If {n_{k}} is a strongly good universal sequence in L_{p}, then the bounded {n_{k}}-Besicovitch sequences are good weight sequences along {n_{k}} for all Dunford-Schwartz operators on L_{p}.

Proof. We only sketch the proof. More details of the argument can be found in the proof of Theorem 1.2 in [12]. By (the proof of) Theorem 4.1 in [11], the constant sequence {a(k) = 1} is a good weight sequence along any strongly good universal sequence, for Dunford-Schwartz operators in L_{p} (p is fixed). Thus the constant sequence is a good weight sequence along {n_{k}} for operators of the form λT, where λ is a complex number with |λ| = 1, since these operators are Dunford-Schwartz as well. We then have convergence a.e. for the averages given by (1) when {a(k)} is a trigonometric polynomial.

Let {a(k)} be a bounded {n_{k}}-Besicovitch sequence. Fix f ∈ L_{∞}. If {b(k)} is a trigonometric polynomial with ∥a(k) − b(k)∥_{1,{n_{k}}} < ϵ, then we have a.e.

lim sup_{N→∞} |\frac{1}{N} ∑_{k=1}^{N} a(n_{k})T^{n_{k}}f − \frac{1}{N} ∑_{k=1}^{N} b(n_{k})T^{n_{k}}f| ≤ ϵ∥f∥_{∞}.

Since ϵ > 0 is arbitrary, we obtain a.e. convergence of the weighted averages given by (1) for f bounded, using the convergence for trigonometric polynomials. Since {a(k)} is bounded and {n_{k}} is good universal for L_{p}, we have for f ∈ L_{p}

| sup_{N} \frac{1}{N} ∑_{k=1}^{N} a(n_{k})T^{n_{k}}f| ≤ ∥\{a(k)\}∥_{∞} sup_{N} \frac{1}{N} ∑_{k=1}^{N} |T |^{n_{k}}|f|

which is finite a.e. An application of the Banach principle completes the proof. □

Remarks. 1. The proof required only that {n_{k}} be good universal in L_{p}, with the constant sequence {a(k) = 1} a good weight along {n_{k}} for all Dunford-Schwartz operators on L_{p}. For p = 1 and {n_{k} = k}, this is satisfied with {n_{k}} not strongly good universal.

2. A. Bellow [2] proved that for any fixed p > 1, there are subsequences {n_{k}} which are good universal sequences in L_{p}, but not in L_{r} with 1 ≤ r < p.

3. In Theorem 4.1 of [11], the following lemma is implicitly applied to sequences which are good universal in L_{p} for every 1 < p < ∞.

Lemma 2.2. Let {n_{k}} be a good universal sequence in L_{p}, for all p in an open interval (r, s), 1 ≤ r < s. Then {n_{k}} is a strongly good universal sequence in L_{p} for every p ∈ (r, s).

Proof. Let τ be an ergodic measure preserving transformation. By Sawyer’s theorem ([18] or [10]), the a.e. convergence of the averages \frac{1}{N} ∑_{k=1}^{N} f(τ^{n_{k}}x) for every f ∈ L_{p} implies that the corresponding maximal operator is of weak type (p, p), for every p ∈ (r, s). Let r < p_{1} < p < p_{2} < s. Then this maximal operator is of weak types (p_{1}, p_{1}) and (p_{2}, p_{2}), so by (a special case of) the Marcinkiewicz interpolation theorem [8], it is of strong type (p, p). By Corollary 2.2 of [11], the maximal operator along {n_{k}} of any positively dominated contraction of L_{p}, particularly of any measure preserving transformation, is of strong type (p, p). Since {n_{k}} is good universal, it is strongly good universal. □

Theorem 2.3. Let {n_{k}} be a good universal sequence in L_{p} for all 1 < p < ∞. For a fixed p, if the sequence {a(k) = 1} is a good weight sequence along {n_{k}} for all positive [positively dominated] contractions of L_{p}, then the r, {n_{k}}-Besicovitch sequences with r > p/(p − 1) are good weight sequences along {n_{k}} for positive [positively dominated] contractions of L_{p}.

Proof. Again we only sketch the proof. More details can be found in the proof of Theorem 2.4 of [12]. By the previous lemma, {n_{k}} is strongly good universal in L_{p} for every p, 1 < p < ∞. By Corollary 2.4 of [11], for any positively dominated contraction T of L_{p}, 1 < p < ∞, the maximal operator sup_{N} \frac{1}{N} ∑_{k=1}^{N} |T^{n_{k}}f| is strong type (p, p).

Fix p such that {a(k) = 1} is a good weight sequence along {n_{k}} for all positively dominated contractions of L_{p}, which means we have a.e. convergence of the averages along {n_{k}} for these operators. If T is a positively dominated contraction of L_{p}, so is the operator λT when λ is a complex number of absolute value 1, so we have a.e. convergence of its averages along {n_{k}}, which is convergence in (1) for {a(k) = λ^{k}}.

We now look at the case that {a(k) = 1} is a good weight sequence along {n_{k}} only for positive contractions of L_{p}. Following [17], for T a positive contraction of L_{p}(X) we take the product space of the unit circle with X, and define P [g(z)f(x)] = g(λz)T f(x). Then P extends to a positive contraction of L_{p} of the product space, and applying to P the assumed convergence for positive operators, with g(z) = z and f ∈ L_{p}(X), we obtain a.e. convergence in (1) for {a(k) = λ^{k}}.

We now prove the part of the theorem when {a(k) = 1} is a good weight sequence along {n_{k}} for all positively dominated contractions of L_{p}; the restricted case of positive contractions is obtained by putting S = T in the proof. Let q = p/(p − 1) be the dual index of p, i.e., \frac{1}{p} + \frac{1}{q} = 1, fix r > q, and let T be dominated by a positive contraction S on L_{p}. By [1], there exists a larger L′_{p}, a positive isometric embedding D of L_{p} into L′_{p}, a conditional expectation operator E and a positive invertible isometry Q such that for each n ∈ Z^{+} we have DS^{n}f = EQ^{n}Df. Since Q can be written in the form Q^{n}f(x′) = w_{n}(x′)f(τx′), where τ is a non-singular point transformation, |Q^{n}f(x)|^{s} = R^{n}|f|^{s} where R is an L′_{p/s} isometry for s = r/(r−1) <

p. Since the maximal operator sup_{N} \frac{1}{N} ∑_{k=1}^{N} R^{n_{k}}|f| is strong type (\frac{p}{s}, \frac{p}{s}), we have that the maximal operator sup_{N} \frac{1}{N} ∑_{k=1}^{N}[S^{n_{k}}|f|]^{s} is strong type (p, p) and hence is finite a.e. Thus, if {b_{j}(k)} is a sequence of trigonometric polynomials that approach {a(k)} in the ∥ ▪ ∥_{r,{n_{k}}} semi-norm, then for a.e. x Hölder’s inequality shows that the sequence {a(k)T^{k}f(x)} will converge in the ∥ ▪ ∥_{1,{n_{k}}} semi-norm. □

Corollary 2.4. Fix p, 1 < p < ∞. Let n_{k} = k^{t} (fixed t ∈ N), or let n_{k} denote the k-th prime, then the r, {n_{k}}-Besicovitch sequences, for r > p/(p − 1), are good weight sequences along {n_{k}} for Dunford-Schwartz operators in L_{p} and for positive contractions of L_{p}.

Proof. Recall that Bourgain [6] has proved that the sequence {k^{t}} is strongly good universal in L_{p}, for every p > 1. Wierdl [20] has proved that the sequence of primes is good universal in L_{p}, p > 1. In [13] it is shown that in both cases the constant sequence is a good weight sequence along {n_{k}} for positive contractions of L_{p}, so the previous theorem yields the result for positive contractions of L_{p}. For T Dunford-Schwartz the proof of the previous theorem applies, since a.e. convergence in (1) for {a(k) = λ^{k}} holds by the first part of the proof of Theorem 2.1. □

3. Uniform Sequences

In this section we will consider the uniform sequences of Brunel-Keane [7]. These are bounded Besicovitch sequences with some further restrictions. We will also consider good averaging sequences {n_{k}} such that for every irrational θ ∈ [0, 1), {n_{k}θ} is uniformly distributed (mod 1). We will show that in this case every uniform sequence produced by an apparatus with a connected space is in B_{1,{n_{k}}}. Since a uniform sequence is bounded, this means that those uniform sequences will then also belong to B_{p,{n_{k}}} for all p > 1, and we will be able to apply the results of the previous section to the uniform sequences {a(k)} along the sequences {n_{k}}.

We first give the construction of the uniform sequences of Brunel and Keane [7], the details of which we will need. Let Ω be a compact metric space, B the collection of Borel subsets of Ω, and ϕ a homeomorphism of Ω such that {ϕ^{n}}_{n≥0} is an equicontinuous family of mappings. The system (Ω, ϕ) is then called uniformly L stable. We assume that Ω possesses a dense orbit. It then follows (see [7] or [17]) that there exists a unique ϕ invariant probability measure on (Ω, B), denoted by ν. Then for any w ∈ Ω, and any continuous function f on Ω,

lim_{n} \frac{1}{n} ∑_{t=0}^{n−1} f(ϕ^{t}w) = ∫ f dν .

Such a system (Ω, B, ν, ϕ) is called strictly L stable.

If (Ω, B, ν, ϕ) is strictly L stable, Y ∈ B with ν(Y) > 0, ν(∂Y) = 0 and y ∈ Ω, the sequence {a_{k}(y)} = {X_{Y} (ϕ^{k}y)} is called a uniform sequence of weights. The entire collection {(Ω, B, ν, ϕ), y, Y } is called the apparatus producing the uniform sequence of weights. The apparatus is said to be connected if Ω is connected. It is clear that a uniform sequence is a “return times” sequence. In fact, we will be interested in the sequences {a_{k}(y)} for all y ∈ Ω. This should be contrasted with the usual situation for return times weights, where one considers only y in a subset Ω′ of Ω, where ν(Ω′) = 1. The fact that uniform sequences of weights are bounded Besicovitch is proved in [17], p. 149. V. Losert has shown us (private communication) that uniform sequences need not be weakly almost periodic.

Theorem 3.1. Let {n_{k}} be a good universal sequence in L_{p}, p > 1 fixed. For θ ∈ [0, 1) irrational such that {n_{k}θ} is uniformly distributed mod 1, let T be induced by the measure preserving transformation τx = θ+x mod 1. Then for all f ∈ L_{p}[0, 1), we have

lim_{N→∞} \frac{1}{N} ∑_{k=1}^{N} T^{n_{k}}f = ∫ f a.e. (2)

Proof. We first note that if (2) holds for a dense class of functions in L_{p}, then we are done: given ϵ > 0 and f ∈ L_{p}[0, 1) we can choose f′ in our dense class such

that ∥f − f′∥_{p} < ϵ. Putting A_{N} = \frac{1}{N} ∑_{k=1}^{N} T^{n_{k}}, we then have

∥A_{N}f − ∫ f∥_{p} ≤ ∥A_{N}f − A_{N}(f′) + A_{N}(f′) − ∫ f′ + ∫ f′ − ∫ f∥_{p}

≤ \frac{ϵ}{3} + \frac{ϵ}{3} + \frac{ϵ}{3}.

Thus ∥A_{N}f − ∫ f∥_{p} → 0. Since the a.e. convergence of the sequence {A_{N}f} is assumed, the limit must be ∫ f.

To see the dense class, we just note that for characteristic functions of intervals, by the assumption that (n_{k}θ) is uniformly distributed, we have convergence to the integral. Hence it is true for finite linear combinations of characteristic functions of intervals, and these are dense. □

Definition. Let τ be a measure preserving point transformation. We say that τ is totally ergodic if the transformations τ^{n}, n = 1, 2, ... are all ergodic.

We can now extend the previous theorem to totally ergodic transformations as opposed to irrational rotations of the circle.

Theorem 3.2. Let {n_{k}} be a good universal sequence in L_{p}, for every 1 < p < ∞, such that {n_{k}θ} is uniformly distributed mod 1 for all θ ∈ [0, 1) irrational, τ a totally ergodic measure preserving point transformation of a probability space X, f ∈ L_{p}(X), p > 1. Then for a.e. x we have

lim_{N→∞} \frac{1}{N} ∑_{k=1}^{N} f(τ^{n_{k}}x) = ∫ f.

Proof. Let λ = e^{2πiθ} be a complex number of modulus one that is not a root of unity. By Theorem 3.1, for the function f(z) = z defined on the unit circle, we have for a.e. z

lim_{N→∞} \frac{1}{N} ∑_{k=1}^{N} λ^{n_{k}}z = ∫_{\{z:|z|=1\}} z = 0.

Hence

lim_{N→∞} \frac{1}{N} ∑_{k=1}^{N} λ^{n_{k}} = 0.

We know that for f ∈ L_{p}, p > 1,

lim_{N→∞} \frac{1}{N} ∑_{k=1}^{N} f(τ^{n_{k}}x)

exists for a.e. x, and hence also in L_{p} norm. For f ∈ L_{2}, we have

\frac{1}{N} ∑_{k=1}^{N} (f ∘ τ^{n_{k}}, f) = \frac{1}{N} ∑_{k=1}^{N} ∫_{\{λ:|λ|=1\}} λ^{n_{k}}dE_{f}(λ)

where dE_{f}(λ) is the spectral measure of the linear operator on L_{2} defined by T f = f ∘ τ. We have shown that

lim_{N→∞} \frac{1}{N} ∑_{k=1}^{N} λ^{n_{k}} = 0

unless λ belongs to the countable set of the roots of unity. But since τ is totally ergodic, no root of unity ≠ 1 is an eigenvalue of T on L_{2}. Hence, no root of unity except 1 is an atom of the spectral measure of T, so for f ∈ L_{2},

lim_{N→∞} \frac{1}{N} ∑_{k=1}^{N} f(τ^{n_{k}}x) = ∫ f .

Since L_{2} ∩ L_{p} is dense in L_{p}, p > 1, the theorem follows as in the proof of Theorem 3.1. □

Even for strongly good universal sequences, the requirement that τ be totally ergodic is necessary. In fact, consider the strongly good universal sequence of the primes. Let λ be a primitive r-th root of unity, and let X be r point space with each point having measure \frac{1}{r}. Let τ be any cyclic permutation of all the points, and let f be defined by f(x) = 1 for one particular x and 0 otherwise. Then it is easy to see that for some x

lim_{N→∞} \frac{1}{N} ∑_{k=1}^{N} f(τ^{n_{k}}x) = 0 ≠ ∫ f

and λ is an eigenvalue for the operator induced by τ, which is ergodic.

Lemma 3.3. If {(Ω, B, ν, ϕ), y, Y } is a connected apparatus producing a uniform sequence, then ϕ is totally ergodic.

Proof. Suppose ϕ is not totally ergodic. Then some power of ϕ has a non-constant invariant function, which implies that the operator S defined by Sf = f ∘ ϕ has an eigenfunction in L_{2} with an associated eigenvalue that is a root of unity.

Let R be the operator S restricted to C(Ω). Then R is almost periodic, so for all λ with |λ| = 1 also \overline{λ}R is almost periodic. Consequently, the averages

\frac{1}{N} ∑_{k=1}^{N} \overline{λ}^{k}R^{k}f

converge uniformly to the projection of f onto the eigenspace of R associated with λ.

If this projection is non-zero, which will happen if and only if λ is an eigenvalue for S (since C(Ω) is dense in L_{2}(Ω) ), R (and hence S) will have a continuous eigenfunction g associated with the eigenvalue λ. Then g assumes only the values g(x_{0}), g(ϕx_{0}), …, g(ϕ^{r−1}x_{0}), where x_{0} has a dense orbit. Since g is continuous and Ω is connected, this is a contradiction. □

Theorem 3.4. Let {n_{k}} be a good universal sequence in L_{∞}, such that for every irrational θ ∈ [0, 1) the sequence {n_{k}θ} is uniformly distributed mod 1, and let {a(k)} be a uniform sequence produced by a connected apparatus {(Ω, B, ν, ϕ), y, Y }. Then {a(k)} is {n_{k}}-Besicovitch.

Proof. By Lemma 3.3, ϕ is totally ergodic. Let g ∈ C(Ω). Since {n_{k}} is a good universal sequence, we have from Theorem 3.2

lim_{N→∞} \frac{1}{N} ∑_{k=1}^{N} g(ϕ^{n_{k}}y) = ∫ g (3)

for a.e. y ∈ Ω. Since g is uniformly continuous, {ϕ^{n}}_{n≥0} is an equicontinuous family, and open sets have positive measure, we have that (3) holds for all y ∈ Ω.

Let g_{1} and g_{2} be continuous functions such that g_{1}(y) ≤ X_{Y} (y) ≤ g_{2}(y) for all y ∈ Ω and ∫ g_{2} − ∫ g_{1} < ϵ, where ϵ > 0 is arbitrary (see [7, 17]). We then have, for all y ∈ Ω,

lim_{N→∞} \frac{1}{N} ∑_{k=1}^{N} g_{1}(ϕ^{n_{k}}y) ≤ lim inf_{N→∞} \frac{1}{N} ∑_{k=1}^{N} a(n_{k})

≤ lim sup_{N→∞} \frac{1}{N} ∑_{k=1}^{N} a(n_{k}) ≤ lim_{N→∞} \frac{1}{N} ∑_{k=1}^{N} g_{2}(ϕ^{n_{k}}y)

Fix y. Then {g_{2}(ϕ^{n}y)} is almost periodic, and using (3) we obtain

lim sup_{N→∞} \frac{1}{N} ∑_{k=1}^{N} |g_{2}(ϕ^{n_{k}}y) − a(n_{k})| ≤

lim_{N→∞} \frac{1}{N} ∑_{k=1}^{N}[g_{2}(ϕ^{n_{k}}y) − g_{1}(ϕ^{n_{k}}y)] = ∫ (g_{2} − g_{1}) < ϵ.

Since ϵ is arbitrary, {a(k)} is {n_{k}}-Besicovitch.

Furthermore, for any ϵ > 0 we also have

lim sup_{N→∞} \frac{1}{N} ∑_{k=1}^{N} a(n_{k}) − lim inf_{N→∞} \frac{1}{N} ∑_{k=1}^{N} a(n_{k}) < ϵ .

Since ϵ is arbitrary, lim_{N→∞} \frac{1}{N} ∑_{k=1}^{N} a(n_{k}) exists (and equals ν(Y) ). □

Remark. To get a better picture of the class of sequences {a(k)} considered in the theorem, we note that Halmos and von-Neumann proved (see Theorem 3 of [17]) that every strictly L stable system is isomorphic to a rotation by a generator of a compact metric monothetic group. Thus, we may assume that Ω is a compact metric (connected) monothetic group, ν its Haar measure, and ϕ(x) = x + α with {α^{n}} dense in Ω. The referee has remarked that this yields an alternative proof of Lemma 3.3.

Combining the previous theorem with the results of the previous section, we obtain the following corollaries.

Corollary 3.5. Fix p, 1 ≤ p < ∞. Let {n_{k}} be a strongly good universal sequence in L_{p}, such that {n_{k}θ} is uniformly distributed mod 1 for every irrational θ ∈ [0, 1). Then any uniform sequence {a(k)} produced by a connected apparatus is a good weight sequence along {n_{k}} for Dunford-Schwartz operators in L_{p}.

Corollary 3.6. Let {n_{k}} be a good universal sequence in L_{s} for every 1 < s < ∞, such that {n_{k}θ} is uniformly distributed mod 1 for every irrational θ ∈ [0, 1). If for a fixed p the constant sequences are good weight sequences along {n_{k}} for positive [positively dominated] contractions of L_{p}, then any uniform sequence {a(k)} produced by a connected apparatus is a good weight sequence along {n_{k}} for positive [positively dominated] contractions of L_{p}.

Corollary 3.7. If n_{k} = k^{t} (for fixed t ∈ N), or if n_{k} denotes the k-th prime, and T is a Dundord-Schwartz operator or a positive contraction of L_{p}(X), p > 1, then for any uniform sequence {a(k)} produced by a connected apparatus and for all f ∈ L_{p}(X), we have

lim_{n→∞} \frac{1}{N} ∑_{k=1}^{N} a(n_{k})T^{n_{k}}f

exists a.e.

Proof. Bourgain [6] has established that for fixed t ∈ N, {n_{k} = k^{t}} is a strongly good universal sequence in L_{p} for all p > 1. Weyl’s theorem ([14], p. 27) says that for θ irrational, {k^{t}θ} is uniformly distributed mod 1. For {n_{k}} the sequence of primes, Wierdl [20] has established that it is good universal in L_{p} for every p > 1, and the uniform distribution of {n_{k}θ} for irrational θ follows from [9], Theorem 9.8. Hence the hypotheses of Theorem 3.4 are satisfied in both cases, and Corollary 2.4 yields the result. □

References

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98 Roger L. Jones, Michael Lin, and James Olsen

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Roger L. Jones: Department of Mathematics, DePaul University, 2219 N. Kenmore, Chicago, IL 60614

rjones@condor.depaul.edu http://www.depaul.edu/˜ rjones/

Michael Lin: Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel

lin@math.bgu.ac.il

James Olsen: Department of Mathematics, North Dakota State University, Fargo, N.D. 58105

jolsen@plains.nodak.edu

This paper is available via http://nyjm.albany.edu:8000/j/1998/3A-7.html.