WIENER'S LEMMA ALONG PRIMES AND OTHER SUBSEQUENCES CHRISTOPHE CUNY, TANJA EISNER, AND BA´ LINT FARKAS arXiv:1701.00101v4 [math.FA] 9 Aug 2017 Abstract. Inspired by subsequential ergodic theorems we study the validity of Wiener's lemma and the extremal behavior of a measure µ on the unit circle via the behavior of its Fourier coefficients µ(kn) along subsequences (kn). We focus on arithmetic subsequences such as polynomials, primes and polynomials of primes, and also discuss connections to rigidity and return times sequences as well as measures on R. We also present consequences for orbits of operators and of C0-semigroups on Hilbert and Banach spaces extending the results of Goldstein [31] and Goldstein, Nagy [33]. 1. Introduction Wiener's lemma is a classical result connecting the asymptotic behavior of the Fourier coefficients µ(n) = zndµ(z) T of a complex Borel measure µ on the unit circle T with its values on singletons. Despite its elementary proof, it has found remarkable applications in several areas of mathematics such as ergodic theory, operator theory, group theory and number theory. Theorem 1.1 (Wiener's Lemma). Let µ be a complex Borel measure1 on the unit circle T. Then lim N 1 N N |µ(n)|2 n=1 = a |µ({a})|2. atom (Since µ(-n) = µ(n), one can replace here 1 N N n=1 by 1 2N +1 N n=-N .) As a consequence, one has the following characterization of Dirac measures in terms of their Fourier coefficients. Here we restrict ourselves to probability measures and give the proof for the reader's convenience. Corollary 1.2 (Extremal behavior of Dirac measures). For a Borel probability measure µ on T the following assertions are equivalent: (i) lim N 1 N N n=1 |µ(n)|2 = 1. (ii) lim |µ(n)| = 1. n (iii) µ is a Dirac measure. 2010 Mathematics Subject Classification. 43A05, 43A25, 47A10, 47B15, 47A35, 37A30, 47D06. Key words and phrases. Wiener's lemma for subsequences, extremal measures, polynomials and primes, ergodic theorems, orbits of operators and operator semigroups. 1By definition finite. 1 2 CHRISTOPHE CUNY, TANJA EISNER, AND BA´ LINT FARKAS Note that by the Koopman­von Neumann lemma, see Lemma 2.1 (b) below and the paragraph preceding it, (i) is equivalent to |µ(n)| 1 in density as n i.e., to the existence of a subset J of density 1 with limn,nJ |µ(n)| = 1. Proof. Let µ be a probability measure on T. If µ({a}) = 1 for some a T, then µ(n) = an. Whence the implication (iii)(ii) follows, while (ii)(i) is trivial. To show (i)(iii), suppose (i). Theorem 1.1 yields 1 = lim N 1 N N |µ(n)|2 n=1 = a µ({a})2 atom a µ({a}) atom = 1, implying µ({a})2 = µ({a}) or, equivalently, µ({a}) {0, 1} for every atom a. We conclude that µ is a Dirac measure. The previous two results have the following operator theoretic counterparts. Theorem 1.3 (Goldstein [32], Ballotti, Goldstein [5]). Let T be a (linear) contraction on a Hilbert space H, and for C denote by P the orthogonal projection onto ker( - T ). Then for every x, y H lim N 1 N N |(T nx|y)|2 n=1 = |(Px|Py)|2 T = |(Px|y)|2. T It is easy to deduce the previous theorem from Wiener's lemma (and vice versa), even though the original proof of Goldstein went along different lines. The following Banach space version of Corollary 1.2 is more complex, see also Lin [47] and Baillon, Guerre-Delabri`ere [4] for related results. Note that in these papers, the results are formulated for C0-semigroups but are also valid for powers of operators with analogous proofs. Theorem 1.4 (Goldstein, Nagy [33]). Let T be a (linear) contraction on a Banach space E. Suppose for some x E | T nx, x | | x, x | for every x E as n . Then ( - T )x = 0 for some T. The aim of this paper is to study the validity of Wiener's Lemma, Corollary 1.2 and Theorems 1.3, 1.4 along subsequences of N, where we study the equivalences (i)(iii) and (ii)(iii) in Corollary 1.2 separately. First of all, some words about terminology. The term complex measure refers to C-valued -additive set function (which is then automatically finite valued, and has finite variation). In this paper only Borel measures will be considered. A subsequence (kn) in N will refer to a function k : N N which is strictly increasing for sufficiently large indices. Banach and Hilbert spaces will be considered over the complex field C. Sequences for which Wiener's lemma and the extremality of Dirac measures work well include certain polynomial sequences, the primes, certain polynomials of primes and certain return times sequences as will be shown below. As an application of the general results we shall prove among others the following, maybe at first glance surprisingly looking, facts. We denote by pn the nth prime. 1) The only Borel probability measures on T with |µ(pn)| 1 for n are the Dirac measures (Theorem 4.4). WIENER'S LEMMA ALONG SUBSEQUENCES 3 2) If T is a (linear) contraction on a Hilbert space and x H \ {0} is such that |(T pnx|x)| x 2 as n , then x is an eigenvector of T to a unimodular eigenvalue (Theorems 4.4 and 5.9). 3) If T is a power bounded operator on a Banach space E and x E \ {0} is such that | T pn x, x | | x, x | as n for every x E, then x is an eigenvector of T to a unimodular eigenvalue (Corollary 5.6). We also relate our results to rigidity sequences and discover a property of such sequences as a byproduct which appears to be new. Our results are inspired by ergodic theory, where the study of ergodic theorems along subsequences has been a rich area of research with connections to harmonic analysis and number theory. Furstenberg [30] described norm convergence of ergodic averages of unitary operators along polynomials. Pointwise convergence of ergodic averages for measure preserving transformations along polynomials and primes, answering a question of Bellow and Furstenberg, was proved by Bourgain [12, 13, 14] and Wierdl [60], with polynomials of primes treated by Wierdl [59] and Nair [52, 51]. To illustrate the wealth of literature on ergodic theorems along subsequences we refer, e.g., to Bellow [8], Bellow, Losert [7], Baxter, Olsen [6], Rosenblatt, Wierdl [56], Berend, Lin, Rosenblatt, Tempelman [9], Boshernitzan, Kolesnik, Quas, Wierdl [11], Krause [42], Zorin-Kranich [64], Mirek [48], Eisner [19], Frantzikinakis, Host, Kra [29], Wooley, Ziegler [62]. The paper is organized as follows. Section 2 is devoted to an abstract version of Wiener's lemma along subsequences. In Section 3 we study extremal and Wiener extremal subsequences, see Definition 3.1. The case of polynomials, primes and polynomials of primes is treated in Section 4. Section 5 is devoted to applications to orbits of operators on Hilbert and Banach spaces. The continuous parameter case is discussed in Section 6, where parallels and differences to the time discrete case are pointed out. Acknowledgment. The authors thank Michael Lin, Rainer Nagel and J´anos Pintz for helpful comments and references. 2. Wiener's Lemma along subsequences Recall that a sequence (an) in C is called convergent in density to a C, with notation D- limn an = a if there exists a set J N of density 1 with limn,nJ an = a. The density of a set J N is defined by limn |J {1,...,n}| n , provided the limit exists. The following is the classical Koopman­von Neumann lemma together with a slight variation. Lemma 2.1. (a) For a bounded sequence (an) in [0, ) the following are equiva- lent: (i) D- limn an = 0. (ii) limN 1 N N n=1 an = 0. (iii) limN 1 N N n=1 a2n = 0. (b) For a bounded sequence (bn) in (-, 1] the following are equivalent: (i) D- limn bn = 1. (ii) limN 1 N N n=1 bn = 1. If bn 0, then these assertions are also equivalent to: (iii) limN 1 N N n=1 b2n = 1. 4 CHRISTOPHE CUNY, TANJA EISNER, AND BA´ LINT FARKAS Proof. (a), (i)(ii) is the content of the Koopman­von Neumann lemma, see, e.g., [40] or [21, Ch. 9], whereas (i)(iii) is a direct consequence. (b) follows from (a) by considering an := 1 - bn. We further recall the following notion from Rosenblatt, Wierdl [56], see also [21, Chapter 21]. Definition 2.2. A subsequence (kn) of N is called good if for every T the limit lim 1 N kn =: c() exists. N N n=1 Moreover, (kn) is called an ergodic sequence if c = 1{1}, the characteristic function of {1}. We call the set := { : c() = 0} the spectrum of the sequence (kn) in analogy to, e.g., Lin, Olsen, Tempelman [46]. By an application of the spectral theorem it follows that a sequence (kn) is good if and only if it is good for the mean ergodic theorem, that is if for every measure preserving system (X, , T ) and every f L2(X, ) the averages 1 N T kn f N n=1 converge in L2(X, ), where T denotes the Koopman operator corresponding to the transformation T defined by f f T . A good sequence is then ergodic if and only if the above limit always equals the orthogonal projection PFix(T )f onto the fixed space Fix(T ) = ker(1 - T ). Remark 2.3. For each subsequence (kn) of N, the sequence (kn ) is equidistributed in T for almost every T implying that c() exists and is 0 for Lebesgue almost every T, see, e.g., Kuipers, Niederreiter [43, Theorem 1.4.1] (or Theorem 2.2 on page 50 of [56], or combine Kronecker's lemma with Carleson's theorem). The function c clearly satisfies c(1) = 1, c() = c() and |c()| 1 whenever c() exists. Moreover, if |c()| = 1, then kn converges to c() in density, which follows by Lemma 2.1 (b) applied to an := Re(c()kn ). Thus c is a multiplicative function on the subgroup { : |c()| = 1} of T. The function c : T C is Borel measurable (if it exists). We present one more property of the limit function c. For an integer d 0 we set Gd := { T : d = 1}, the group of dth roots of unity. Proposition 2.4. Let (kn) be a good sequence with corresponding limit function c. Then there exists an integer d 0 such that := { T : |c()| = 1} = Gd. Proof. It follows from Remark 2.3 that is a group, and it is then well-known that is either finite or dense in T. We shall prove that it is finite. Since (kn) is good, c is the pointwise limit of a sequence of continuous functions on a compact space. Hence, by a theorem of Baire, its set of continuity points is dense in T. As mentioned in Remark 2.3, limN 1 N N n=1 kn = 0 for almost every (with respect to the Haar measure on T). If is not finite, we infer that c is nowhere continuous, which is impossible. The following general fact may appear to be well known, but we could not find a reference. WIENER'S LEMMA ALONG SUBSEQUENCES 5 Proposition 2.5 (Wiener's lemma along subsequences). Let (kn) be a good sequence in N. (a) For every complex Borel measure µ on T lim N 1 N N |µ(kn)|2 n=1 = c(12)d(µ × µ)(1, 2). T2 (b) The sequence (kn) is ergodic if and only if lim N 1 N N |µ(kn)|2 n=1 = a |µ({a})|2 atom holds for every complex Borel measure µ on T. (c) For an ergodic sequence (kn) and a Borel probability measure µ on T the limit above in (b) is 1 if and only if µ is a Dirac measure. Proof. (a) The proof goes along the same lines as the most elementary and wellknown proof of the Wiener lemma. Observe that, by Fubini's theorem and by Lebesgue's dominated convergence theorem, 1 N N |µ(kn)|2 = 1 N N n=1 n=1 k1n dµ(1) T 2kn dµ(2) T = T×T 1 N N (12)kn d(µ n=1 × µ)(1, 2) c(12)d(µ × µ) as N . T2 (b) If now c = 1{1} we see that, by Fubini' theorem, the limit above equals 1{1}(2)dµ(1) dµ(2) = µ({2})dµ(2) = |µ({a})|2. TT T a atom For the converse implication suppose (kn) is not ergodic, and let T\{1} be with c() = 0. If Re c() = 0, then consider the probability measure µ := 1 2 (1 + ). We then have 11 T2 c(12)d(µ × µ)(1, 2) = 2 + 4 c() + c() = 1 2 = a |µ({a})|2. atom If Im c() = 0, then for the measure µ := 1 2 (1 + i) we have T2 c(12)d(µ × µ)(1, 2) = 1 2 + i 4 c() - c() = 1 2 = a |µ({a})|2. atom The proof of (b) is complete. (c) follows from (b) by a similar arguments as in the proof of Corollary 1.2. The following questions arise naturally, cf. also Proposition 3.15 below. Question 2.6. Does the existence of limN 1 N N n=1 |µ(kn )|2 for every proba- bility Borel measure µ implies that (kn) is good? Is there a non-ergodic, good sequence (kn) with Re c() = 0 for each T \ {1}? 6 CHRISTOPHE CUNY, TANJA EISNER, AND BA´ LINT FARKAS Remark 2.7. If one replaces "probability measure" by "complex measure", the an- swer to the first question is positive. Indeed, it is easy to see that for a subsequence (kn)nN of N the following assertions are equivalent. (i) For every finite complex measure (respectively every probability measure) µ on T the limit limN 1 N N n=1 |µ^(kn)|2 exists; (ii) For every complex measure (respectively every probability measure) µ and on T the limit limN 1 N N n=1 Re(µ^(kn)^(kn)) exists. Assume that the above equivalent conditions hold for probabilities. Taking := 1 and µ := , we see that the limit limN 1 N N n=1 Re(kn ) exists. If we assume moreover that the conditions hold for finite complex measures, then, taking := i1 and µ := , we see that the limit limN 1 N N n=1 Im(kn ) exists, implying that (kn) is good. Corollary 2.8. Let (kn) be a good sequence. For a Borel probability measure µ lim sup N 1 N N |µ^(kn)|2 n=1 = 1 (1) holds if and only if µ is discrete with c(ab) = 1 for all atoms a, b. (2) In this case, the limit superior is a limit, and µ is supported in a coset Gd for some integer d 0. Proof. Suppose (1) holds. Since (kn) is good, by Proposition 2.5 (a) the above limit superior is actually a limit and 1 N N |µ^(kn)|2 n=1 - N c(1Ż2)dµ(1)dµ(2) = 1. T×T Hence 1 - Re(c(1Ż2)) dµ(1)dµ(2) = 0. T×T Since 1 - Re(c(1Ż2)) 0 (and |c| 1). We infer that there exists 2 T such that for µ-a.e. 1 T, c(1Ż2) = 1. Hence, µ is supported on 2, which equals 2Gd for some integer d 0 by Proposition 2.4. This shows one implication. For the converse implication let µ be discrete satisfying (2). Then by Proposition 2.5 (a) 1 N N |µ^(kn)|2 = c(ab)µ({a})µ({b}) = µ({a})µ({b}) = 1. n=1 a,b atom a,b atom We now consider the case "in between", namely when c() = 0 for all but at most countably many 's. We first introduce the following terminology: For a subset of T denote by the subgroup generated . We call two elements 1, 2 T -dependent if their cosets with respect to coincide: 1 = 2 , otherwise we call them -independent. Theorem 2.9. Let (kn) be a good sequence with at most countable spectrum . WIENER'S LEMMA ALONG SUBSEQUENCES 7 (a) For every complex Borel measure µ on T lim N 1 N N |µ(kn)|2 n=1 = c() a µ({a})µ({a}). atom In particular, for every continuous complex Borel measure µ on T lim N 1 N N |µ(kn)|2 = 0. n=1 (b) For every Borel probability measure µ on T lim N 1 N N |µ(kn)|2 n=1 µ(a aU )2, (3) where U is a maximal set of -independent atoms. The equality in (3) holds if and only if µ satisfies (2) (but it may not necessarily be discrete). Proof. (a) By Proposition 2.5 and Fubini's theorem we have lim N 1 N N |µ(kn)|2 = n=1 c(12)d(µ × µ)(1, 2) T2 = c()µ({1})dµ(1) T = c() µ({1})dµ(1) T = c() µ({a})µ({a}). a atom (b) Observe (the left-hand side below is greater or equal to zero by (a)) c() µ({a})µ({a}) µ({a}) c()µ({a}) a atom a atom µ({a}) µ({a}) a atom = µ({a})µ(a ) a atom = µ(a )2. aU The last assertion regarding the equality is clear. We will see below that there are sequences (kn) satisfying the assumptions of Theorem 2.9 and probability measures satisfying (1) which are not Dirac. Remark 2.10. Let (kn) be a strictly increasing sequence in N having positive density. If the characteristic function of {kn : n N} is a Hartmann sequence (i.e., has Fourier coefficients), then (kn) is good with at most countable spectrum (see, e.g., Lin, Olsen, Tempelman [46] or Kahane [39]). 8 CHRISTOPHE CUNY, TANJA EISNER, AND BA´ LINT FARKAS By using a result2 of Boshernitzan, which we recall for the sake of completeness, it is possible to show that good sequences with positive upper density have countable spectrum. Actually, his Theorem 41 in Rosenblatt [55] is stated in the case where nk = k for every k N, but the proof is the same. Proposition 2.11 (Boshernitzan, see Rosenblatt [55]). Let (an) be a bounded sequence of complex numbers and let (Nm) be a subsequence of N. For every > 0, the set T : lim inf k 1 Nm Nm a =1 is finite. Recall that the upper density of a subsequence (kn) of N is defined by d(kn) := lim sup N |{n : kn N N }| . Corollary 2.12. Let (kn) be a good subsequence of N with positive upper density. For every > 0 the set { T : |c()| } is finite. In particular, (kn) has countable spectrum. Proof. By assumption, there exists a subsequence (Nm)mN of N such that lim m |{n : kn Nm Nm}| = > 0. Then, for A = {kn : n N} and for every T we obtain 1 Nm Nm =1 1A() - m c(). An application of Proposition 2.11 with a = 1A() finishes the proof. Remark 2.13. A good sequence need not have positive upper density as, e.g., kn = n2 shows. See Section 4 below for this and other examples. On the other hand, a sequence with positive upper density (and even density) does not have to be good. Indeed, take 2N and change 2n to 2n + 1 if 2n lies in any interval of the form [4, 2·4], N. This sequence has density 1/2 but c(-1) does not exist. Modifying this construction it is easy to construct a sequence with density arbitrarily close to 1 which is not good. (Note that 1 cannot be achieved: every sequence with density 1 is automatically good.) Remark 2.14. Suppose (kn) is a subsequence of N (not necessarily good) such that there is an at most countable set such that c() exists and equals 0 for every . By carrying out the same calculation as in the proof of (a) in Proposition 2.5 and using the Koopman­von Neumann Lemma 2.1 we see that for each continuous measure on T we have µ(kn) 0 in density. It would be interesting to characterize those subsequences (kn) for which a (probability) measure is continuous if and only if µ(kn) 0 in density. 2We thank Michael Lin for bringing the reference [55] to our attention. WIENER'S LEMMA ALONG SUBSEQUENCES 9 3. (Wiener) extremal subsequences In this section we characterize subsequences (kn) for which the equivalences (i)(iii) and (ii)(iii) in Corollary 1.2 remain valid and show that (i)(ii) fails in general. Definition 3.1. Let (kn) be a subsequence of N. We call a Borel probability measure µ Wiener extremal or extremal along (kn) if µ satisfies lim N 1 N N |µ(kn)|2 = 1 or lim n |µ(kn)| = 1, respectively. n=1 A subsequence (kn) in N is called (Wiener) extremal if every (Wiener) extremal measure is a Dirac measure. If every (Wiener) extremal discrete measure is Dirac, then we call (kn) (Wiener) extremal for discrete measures. We first consider Wiener extremal sequences. Theorem 3.2. For a subsequence (kn) of N consider the following assertions: (i) (kn) is Wiener extremal. (ii) (kn) is Wiener extremal for discrete measures. (iii) For each z T whenever D- lim zkn 1, n then z = 1. (iv) c() = 1 implies = 1. Then (i)(ii)(iii)(iv). Moreover, (i)(ii) if (kn) is good. Proof. (i)(ii) is trivial and (iii)(iv) follows from Remark 2.3. (iii)(ii): Assume that there exists a discrete probability measure which is extremal and not Dirac. Let a, b be two different atoms of µ. Since |µ(n)| |anµ({a}) + bnµ({b})| + µ({}) 1, =a,b atom the extremality of µ implies that |akn µ({a}) + bkn µ({b})| converges in density to µ({a}) + µ({b}) or, equivalently, that |akn - bkn | converges in density to 1. Taking z := ab = 1 in (iii), we arrive at a contradiction. (ii)(iii): Assume that there exists z T with z = 1 such that zkn converges to 1 in density. Then for the probability measure µ defined by µ({1}) = µ({z}) = 1/2 µ(kn) = 1 + zkn 2 converges to 1 in density, hence (i) is false. The last assertion follows immediately from Corollary 2.8. Replacing, in the above proof, the Ces`aro limit by the classical limit and convergence in density by classical convergence yields the following. Theorem 3.3. For a sequence (kn) in N consider the following assertions: (i) (kn) is extremal. (ii) (kn) is extremal for discrete measures. (iii) G((kn)) := {z : zkn 1} = {1}. 10 CHRISTOPHE CUNY, TANJA EISNER, AND BA´ LINT FARKAS Then (i)(ii)(iii). Moreover, (i)(ii) if (kn) is good. Remark 3.4 (Ergodic sequences). By the above characterizations or by Proposition 2.5 (c), every ergodic sequence is Wiener extremal and hence extremal, too. We recall the notation Gd =: { T : d = 1} and observe the following. Proposition 3.5. Let (kn) be a subsequence of N satisfying lim inf n (kn+1 - kn) < . Then any probability measure µ that is extremal along (kn) is discrete with supp(µ) 0Gd for some d N and some 0 T. As a consequence, the following assertions are equivalent: (i) (kn) is extremal. (ii) (kn) is extremal for discrete measures. (iii) For every q N, q 2 there are infinitely many n with kn / qN. Remark 3.6. Note that assertion (iii) above just means that (kn) is extremal for roots of unity, i.e., lim kn = 1, T root of unity = = 1. n Remark 3.7. A sequence (kn) with lim infn(kn+1 - kn) < need not be good. An example is given by the sequence 2, 4, . . . , 2n, . . . where along a subsequence of density 0 we insert 2k + 1 right after 2k; or see Remark 2.13 for a not good sequence with positive density. Conversely, a good sequence (even if Wiener extremal) need not have such small gaps: Again kn = n2 is an example. Also, small gaps in (kn) do not imply that (kn) would be extremal, an example is kn = pn + 1, pn the nth prime. See Section 4 for more information. Proof of Proposition 3.5. By assumption there exists an integer d N and a subsequence (n)N, such that kn+1 - kn = d for all N. (4) Let µ be extremal along (kn), and let n [0, 2) be such that µ^(kn) = ein |µ^(kn)|. Then (1 [0,2) - cos(knt - n))dµ(t) = 1 - |µ^(kn)| - n 0. Hence, (cos(kn · -n ))N admits a subsequence converging µ-a.e. to 1. For simplicity, let us assume that the sequence itself converges µ-a.e. to 1 and that n 0 [0, 2] as . Similarly, we may assume that (cos(kn+1 · -n+1))N converges µ-a.e to 1 and that n+1 1 [0, 2] as . By using (4), we infer that for µ-a.e. t [0, 2) dt - 1 + 0 = 0 mod 2. Hence, µ is a discrete measure with supp(µ) 0Gd for 0 = ei(1-0)/d and the first assertion is proven. By Theorem 3.3, it remains to show (iii)(i). Let µ be extremal along (kn). By the above we have µ = d j=1 cj 0j , where c1, . . . , cd 0 with d j=1 cj = 1, WIENER'S LEMMA ALONG SUBSEQUENCES 11 0 T and 1, . . . , d being the dth roots of unity. The extremality of µ implies |µ^(kn)|2 = d cj kj n 2 = d cj cm(j m)kn - n 1. j=1 j,m=1 By convexity reasons this is possible only if limn(j m)kn = 1 whenever cjcm = 0. Thus (iii) and Remark 3.6 imply j = m whenever cjcm = 0, meaning that µ is Dirac. Remark 3.8. For a subsequence (kn) and a subset J N of density 1, (kn)nJ has the same upper density as (kn)nN by 1 N 1 1 N 1 0 as N . kn N ,n/ J nN,n/J Lemma 3.9. Let (kn)nN be a subsequence of N with positive upper density. Then lim infn(kn+1 - kn) < . Proof. Assume that kn+1 - kn as n . Let A > 0. There exists M > 0 such that for every n M , kn+1 - kn A. Hence, for every n M we have kn kM + (n - M )A (n - M )A. Hence, for every N N large enough, |{n : kn N }| N/A + M and thus d(kn) 1/A 0 as A , resulting in a contradiction. Remark 3.10. It is not difficult to exhibit sequences (kn) with density 0 such that lim infn(kn+1 - kn) < . An important example is the sequence of primes (pn)nN. It is a recent, highly non-trivial result of Zhang that lim infn(pn+1 - pn) < , see [63] or the paper [53] by the Polymath project. We have the following characterization of Wiener extremality for sequences with positive upper density. Note that extremality of such sequences was characterized in Proposition 3.5. Proposition 3.11 (Wiener extremality of sequences with positive upper density). For a subsequence (kn) with positive upper density the following assertions are equivalent: (i) (kn) is Wiener extremal. (ii) (kn) is Wiener extremal for discrete measures. (iii) d({n : kn / qN}) > 0 for every q N, q 2. Note that assertion (iii) above just means that (kn) is Wiener extremal for roots of unity, i.e., D- lim kn = 1, T root of unity = = 1. n Proof. It suffices to show the implications (ii)(i) and (ii)(iii). (ii)(i): Suppose that (kn) is Wiener extremal for discrete measures and let µ be a Wiener extremal measure along (kn) with decomposition µ = µd + µc into discrete and continuous parts. By Lemma 2.1 (b) and Remark 3.8 there exists a subsequence (kn ) of (kn) of positive upper density such that limn |µ^(kn )| = 1. By Theorem 1.1, Lemma 2.1 (a) there is a subsequence (mn) of N of density one 12 CHRISTOPHE CUNY, TANJA EISNER, AND BA´ LINT FARKAS with limn |µ^c(mn)| = 0. Denoting by (n) the non-trivial intersection of (mn) with (kn ) we obtain 1 = lim n |µ^(n)| = lim n |µ^d(n)|. Thus µd is also a probability measure, and therefore µ = µd is Dirac by the as- sumption. (ii)(iii): By the equidistribution of (zn) for any irrational z (i.e., z not a root of unity) combined with Lemma 2.1 (b) and Remark 3.8 we obtain the implication: D- lim zkn = 1 = z is rational. n In particular, by Theorem 3.2 (kn) is not extremal for discrete measures if and only if there exists q N, q 2 such that the set of n with kn qN has density one. Thus, the question of characterizing extremality becomes interesting for sequences of density zero, see Section 4. Example (Return time sequences). Let (X, µ, T ) be an ergodic measure preserving probability system, and let T also denote the corresponding Koopman operator on L2(X, µ) defined by T f = f T . Let A X with µ(A) > 0. We show that for almost every x X the return times sequence (kn) corresponding to {n N : T nx A} is Wiener extremal (and hence extremal) whenever T is totally ergodic. Note that return times sequences play an important role for ergodic theorems, see Bourgain's celebrated return times theorem in Bourgain, Furstenberg, Katznelson, Ornstein [16] and a survey by Assani, Presser [3]. Let A,x(n) := |{k n : T kx A}|. We have for T 1 A,x(n) k kn,T kxA = 1 A,x(n) n 1T -kA(x)k k=1 = n A,x(n) 1 n n (T k1A)(x)k. k=1 Birkhoff's ergodic theorem and the ergodicity assumption imply that for almost every x X lim n A,x(n) n = lim n 1 n n (T k1A)(x) = µ(A), k=1 i.e., the density of (kn) equals µ(A). Hence, by the Wiener­Wintner theorem, see [61], for almost every x c() = 1 µ(A) lim n 1 n n (T k1A)(x)k = 1 µ(A) (P 1A)(x) for all T, k=1 where P denotes the orthogonal projection onto ker( - T ). Thus for almost every x the spectrum of the return times sequence is at most countable. We suppose now that T is totally ergodic and show that (kn) is Wiener extremal. As in the proof of Proposition 3.11, if limN 1 N N n=1 kn = 1, then is rational (i.e., a root of unity). But then total ergodicity implies that c() = 0 for = 1, implying = 1, and this shows that (kn) is Wiener extremal. Note that here total ergodicity cannot be replaced by ergodicity. Indeed, the rotation on two points is ergodic, but for A consisting of one point the return times sequence (kn) = 2N is not extremal. Example (Return time sequences along polynomials). Let (X, µ, T ) be an invertible totally ergodic system, let T denote also its Koopman operator on L2(X, µ) and let µ(A) > 0. Take a polynomial P Z[·] with deg(P ) 2. We show that the return WIENER'S LEMMA ALONG SUBSEQUENCES 13 times sequence (kn) along P corresponding to {n N : T P (n)x A} is ergodic and hence Wiener extremal and extremal for almost every x. (That the sequence is Wiener extremal is also for true for linear polynomials, which can be easily deduced from the previous example.) We let A,x,P (n) := |{k n : T P (k)x A}| and compute for T lim n A,x,P (n) n = lim n 1 n n (T P (k)1A)(x) = µ(A) a.e. x X, (5) k=1 where the last equality follows from a.e. convergence of polynomial averages by Bourgain [12, 13, 14], from the fact that the rational spectrum factor is characteris- tic for polynomial averages (see e.g. Einsiedler, Ward [18, Sec. 7.4]) and from total ergodicity. It is a further result of Bourgain that the limit lim n 1 n n (T P (k)1A)(x)k (6) k=1 exists for each T for a.e. x X, see [24]. Since deg(P ) 2, by the spectral theorem, by the equidistribution of polynomials with at least one irrational nonconstant coefficient and by total ergodicity, the limit in (6) for almost every x X equals µ(A) for = 1 and 0 if = 1. Combining this with (5) gives lim n 1 A,x,P (n) k kn,T P (k)xA = lim n n A,x,P (n) 1 n n (T P (k)1A)(x)k k=1 = 1 if = 1, 0 otherwise, for almost all x X, meaning that (kn) is ergodic for almost all x X. Example (Double return times sequences). Let (X, µ, T ) be a weakly mixing system and let A, B X be with µ(A), µ(B) > 0. We show that the double return times sequence (kn) corresponding to {n N : T nx A, T 2nx B} is for almost every x ergodic and hence Wiener extremal and extremal. By Bourgain [15] the limit lim n 1 n n (T k1A)(x)(T 2k1B)(x) k=1 exists almost everywhere. Moreover, for weakly mixing systems the above limit equals µ(A)µ(B) a.e., see, e.g., [21, Theorem 9.29]. By Assani, Duncan, Moore [2, Theorem 2.3] (or by a product construction), for almost every x, the limit lim n 1 n n (T k1A)(x)(T 2k1B)(x)n k=1 exists for each T and the Host­Kra factor Z2 is characteristic for such averages (meaning that only the projections of 1A and 1B onto this factor contribute to the limit). Since for weakly mixing systems all Host­Kra factors coincide with the fixed factor (see e.g. Kra [41, Sect. 6.1,7.3)]), the above limit equals µ(A)µ(B) for = 1 and to zero otherwise. As before, this shows that the double return times sequence is ergodic for almost every x X 14 CHRISTOPHE CUNY, TANJA EISNER, AND BA´ LINT FARKAS For more ergodic sequences see Boshernitzan, Kolesnik, Quas, Wierdl [11]. Note that since is not yet the pointwise convergence of weighted averages along primes 1 N studied, the return times sequences along primes of the form {n : N n=1 nT T pn x pn A} are currently out of reach. Example (An extremal sequence which is not Wiener extremal). Consider the sequence (kn) defined by the following procedure. Take the sequence (2n)nN and for k belonging to a fixed subsequence of indices with density zero (e.g., the primes) insert 2k + 1 between 2k and 2k + 2. Clearly, (kn) is good with c(1) = c(-1) = 1 and c() = 0 otherwise. Moreover, for z T lim N 1 N N |zkn - 1| = 0 lim N 1 N N |z2n - 1| = 0 z {1, -1}, n=1 n=1 whereas limn |zkn - 1| = 0 is equivalent to z = 1. Thus, by Theorems 3.2 and 3.2, (kn) is extremal but not Wiener extremal. Note that an example of a Wiener extremal measure which is not Dirac is µ given by µ({1}) = µ({-1}) = 1/2. We now go back to Wiener's lemma which in particular implies that a measure µ is continuous if and only if limN 1 N N n=1 |µ(n)|2 = 0. This motivates the following natural question concerning a characterization another kind of extremality for subsequences. Question 3.12. For which subsequences (kn) of N and which continuous measures µ on T does lim N 1 N N |µ(kn)|2 n=1 = 0 (7) hold? For which sequences (kn) does (7) hold for every continuous measure? For which sequences (kn) does (7) characterize continuous measures µ? Remark 3.13. Property (7) characterizes continuous measures for ergodic sequences by Proposition 2.5 (b). Note that by Theorem 2.9, for sequences which are good with at most countable spectrum, (7) holds for all finite continuous measures. The following two examples show however that even for such sequences (7) does not characterize continuous measures in general. Example. Consider (kn) with kn := 2n + 1, which is of course a good sequence with spectrum = {-1, 1} and c(-1) = -1. Let µ = 1 2 (1 + -1). Then we obtain that lim N 1 N N |µ(2n + 1)|2 = 0. n=1 The following observation conjectures a connection between the two kinds of extremality. Remark 3.14. Consider the following assertions about a sequence (kn): (i) (kn) is Wiener extremal for discrete measures and 1 N N n=1 |µ(kn )|2 0 as N for each continuous measure µ. (ii) (kn) is Wiener extremal. WIENER'S LEMMA ALONG SUBSEQUENCES 15 (iii) (kn) is Wiener extremal for discrete measures and 1 N N n=1 |µ(kn )|2 1 as N for each continuous measure µ. (iv) (kn) is Wiener extremal for discrete measures. Then we have the implications (i) (ii) (iii) (iv). Moreover, for good sequences we have also (iv) (ii), i.e., the last three statements are equivalent. Proof. (i) (ii) follows immediately from the decomposition into the discrete and the continuous part and the triangle inequality (note that by the Koopman­von Neumann Lemma 2.1 we can remove the square in (i)), whereas the implications (ii) (iii) (iv) are trivial. The last assertion is Theorem 3.2. We finally discuss connection to rigidity sequences. Recall that for T a sequence (kn) is called a -rigidity sequence if there is a continuous probability measure µ on T with µ(kn) as n . Moreover, 1-rigidity sequences are called rigidity sequences. Note that, although for every T, -rigid (along some subsequence) continuous measures are typical in the Baire category sense in all probability measures, see Nadkarni [50], to check whether a given sequence (kn) is rigid or -rigid is often a challenge. For more details on such sequences, their properties, examples and connections to ergodic and operator theory we refer to Nadkarni [50, Ch. 7], Eisner, Grivaux [23], Bergelson, del Junco, Leman´czyk, Rosenblatt, [10], Aaronson, Hosseini, Leman´czyk [1], Grivaux [36], Fayad, Kanigowski [27], and [20, Section 4.3]. Theorem 2.9 (a) and Corollary 2.8 imply in particular a possibly unexpected necessary property of rigidity sequences. Proposition 3.15. (a) Suppose the sequence (kn) is such that there exists a continuous measure µ on T with lim sup N 1 N N |µ(kn)|2 n=1 > 0. Then either (kn) is not good, or good with uncountable spectrum. (b) -rigidity sequences are not good. For a consequence for prime numbers, polynomials and polynomials of primes see Proposition 4.5 below. Example. The sequence (2n) is a rigidity sequence, see Eisner, Grivaux [23] and Bergelson, del Junco, Leman´czyk, Rosenblatt [10], and, as every lacunary sequence, is not good for the mean ergodic theorem, see Rosenblatt, Wierdl [56, Section II.3]. More examples are sequences satisfying kn|kn+1 or limn kn+1/kn = , although limn kn+1/kn = 1 is possible, for details see the two above mentioned papers, [10] and [23]. 4. Wiener's Lemma along polynomials and primes In this section we consider arithmetic sequences such as values of polynomials, primes and polynomials on primes, inspired by ergodic theorems along such sequences by Bourgain, Wierdl, and Nair, see [12, 13, 14, 60, 59, 52, 51]. Note that all these sequences have density zero (if the degree of the polynomial is greater or equal to two). 16 CHRISTOPHE CUNY, TANJA EISNER, AND BA´ LINT FARKAS The following lemma is classical, see Vinogradov [58], Hua [38], Rhin [54], and Rosenblatt, Wierdl [56, Section II.2]. We present here a quick way to derive it for polynomials of primes from a recent powerful result of Green and Tao [35, Prop. 10.2] on the orthogonality of the modified von Mangoldt function to nilsequences. Lemma 4.1. Let kn = P (n), n N, or kn = P (pn), n N, where P is an integer polynomial and pn denotes the nth prime. Then c() = 0 for every irrational T (not a root of unity). Proof (for polynomials of primes). Let (n) := log n, 0, if n is prime, otherwise, let N and let W = W := p. p prime, p For r < W coprime to W consider the modified -function r, (n) := (W W ) (W n + r), n N, for the Euler totient function . Let now P be an integer polynomial and bn := (n)P (n). Since (P (n))nN can be represented as a Lipschitz nilsequence for a connected, simply connected Lie group, see Green, Tao, Ziegler [34, Appendix C], it follows from Green and Tao [35, Prop. 10.2], see [19, Lemma 3.2 (b), Cor. 2.2], that lim N 1 N N bn = lim 1 W lim N 1 N N bW n+r n=1 r 0 such that (kn) aZ. Then (kn) is not R-extremal and hence not R-Wiener extremal even for discrete measures. Indeed, consider µ := 1 2 (1/a + -1/a). Then µ is not Dirac with µ^(kn) = R e(kn)dµ() = e(kn/a) + e(-kn/a) 2 = 1 for all n N. More generally, any measure in conv{k/a : k Z} provides a similar example. As a corollary, unlike the discrete case, polynomials with rationally dependent coefficients, primes or such polynomials of primes, though being good with count- able spectrum, are not R-extremal and hence not R-Wiener extremal. Note that for such sequences (kn) the periodic unitary group of translations on L2([0, a]) satisfies 30 CHRISTOPHE CUNY, TANJA EISNER, AND BA´ LINT FARKAS T (an) = I for every n Z and thus presents a counterexample to the continuous analogues of (ii) and (iii) in Proposition 5.4, Theorems 5.7, 5.9 and 5.10 and 1)­4) from the introduction. On the other hand, polynomials with rationally dependent non-constant coeffi- cients and rationally independent constant term are R-Wiener extremal and hence R-extremal. Indeed, without loss of generality let kn = P (n) + b for a polynomial P with coefficients from aZ and b being rationally independent from a. Since the spectrum of (kn) is countable, it suffices to show that c() = 1 implies = 0 by a continuous analogue of Theorem 3.2. As in Section 4 one can prove that c() = e(b) lim N 1 N N e n=1 P (n) a a = 1 q q r=1 e(b)e( P (r)d aq ), 0, if a = d q Q, if a / Q. So that c() = 1 implies a = d/q Q for some d Z, q N and (b + P (r))d aZ for all r {1, . . . , q}. Since a, b are rationally independent, d = = 0. Analogously, for such polynomials P the sequence (P (pn)) (pn denoting the nth prime) is RWiener extremal and hence R-extremal, too. Consider finally P R[·] with rationally independent non-constant coefficients. Then for kn := P (n), n N, we have by Weyl's equidistribution theorem c() = 0 for all = 0, i.e., (P (n)) is ergodic. Thus by Proposition 6.6 (c) (P (n)) is R-Wiener extremal and hence R-extremal. Moreover, a suitable modification of Lemma 4.1 using Weyl's equidistribution theorem for polynomials (and the fact that the product of finitely many nilsequences is again a nilsequence) shows that for such polynomials the sequence (P (pn)) is ergodic, and hence R-Wiener extremal. 6.5. Orbits of C0-semigroups revisited. We thus have the following continuous parameter versions of the results from the introduction being the generalizations of the respective results of Goldstein [31] and Goldstein, Nagy [33]. (For the Jacobs­de Leeuw­Glicksberg decomposition for C0-semigroups with relatively compact orbits see, e.g., [20, Theorem I.1.20].) Theorem 6.7. Let (kn) be of the form (P (n)) or (P (pn)), where P R[·] has either rationally independent non-constant coefficients, or rationally dependent nonconstant coefficients which are rationally independent from the constant coefficient, and we suppose that the leading coefficient of P is positive. (a) Let (T (t))t0 be a C0-semigroup of contractions with generator A on a Hilbert space H. Then for any x, y H lim N 1 N N |(T (kn)x|y)|2 n=1 = |(Pax|y)|2, aR where Pa denotes the orthogonal projection onto ker(a - T ). Moreover, lim N 1 N N |(T (kn)x|x)|2 = x4 n=1 for x = 0 implies that x is an eigenvector of A with imaginary eigenvalue. WIENER'S LEMMA ALONG SUBSEQUENCES 31 (b) Let E be a Banach space and (T (t))t0 be a bounded C0-semigroup on E with generator A. Then lim N 1 N N | T (kn)x, x |2 = | x, x |2 for every x E n=1 for x E \ {0} with relatively compact orbit implies that x is an eigenvector of A with imaginary eigenvalue. 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