BERNARD HOST AND BRYNA KRA

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1 UIFORMITY SEMIORMS O l AD A IVERSE THEOREM SUMMARY OF RESULTS BERARD HOST AD BRYA KRA Abstract. For each integer k, we define seminorms on l (Z), analogous to the seminorms defined by the authors on bounded functions in a measure preserving system (associated to the averages in Furstenberg s proof of Szemerédi s Theorem) and to the norms on Z/Z defined by Gowers (in his proof of Szemerédi s Theorem). For the seminorms on l (Z), we prove an inverse theorem. For the Gowers norms on Z/Z, such an inverse theorem was proven for k = 2 and k = 3 by Green and Tao. This is a short summary of some of the results that will eventually be contained in a paper with the above title. Some of these results are related to the analysis in [4].. Seminorms Definition. Let a = (a n : n Z) be a bounded sequence. We say that the averages of the sequence a converge if for all sequences of intervals I = (I j : j ) whose lengths I j tend to infinity, the it j + a n I j exists; by definition, this it does not depend on the sequence of intervals and we denote it by averages (a n ). The upper it of the averages of the sequence a is defined to be sup averages(a n ) = + sup I is an interval of length a n. We use the same definition and notation for sequences a = (a n,...,n k : n,..., n k Z) indexed by Z k ; sequences of intervals whose lengths tend to infinity are replaced by sequences of subsets of Z k of the form R = (R j : j ), where R j = I j, I j,k and I j,,..., I j,k are intervals with min i I j,i + as j +. In fact, we could replace this sequence of sets by any Følner sequence in Z k. otation. We write z Cz for the complex conjugation in C. Thus C k z = z if k is an even integer and C k z = z if k is an odd integer. For ɛ = (ɛ,..., ɛ k ) and h = (h,..., h k ), we define ɛ = ɛ ɛ k and ɛ h = ɛ h ɛ k h k. n I Date: September 8, 2007.

2 2 BERARD HOST AD BRYA KRA Theorem (and definition). Let k be an integer and a be a bounded sequence. We say that a sequence of intervals I = (I i : i ) whose lengths tend to infinity satisfies property (P k ) for the sequence a if for all h = (h,..., h k ) Z k, the it c h (I, a) := i + I i n I i ɛ {0,} k C ɛ a n+h ɛ exists. Let k 2 be an integer and let a be a bounded sequence. If I = (I i : j ) is a sequence of intervals satisfying property (P k ) for a, then the averages of the sequence ( c h (J, a) 2 : h Z k ) converge to a it cor k (J, a). If J = (J j : j ) is a sequence of intervals satisfying property (P k ) for a, then the averages of the sequence (c h (J, a): h Z k ) converge to a it cor k (J, a) 0. We have sup I cor k (I, a) = sup cor k(i, a), J where the upper bounds are taken for all sequences I of intervals satisfying (P k ) and all sequences J of intervals satisfying (P k ) for the sequence a. Writing a U(k) = sup cor k (I, a) /2k, I we have that the map a a U(k) is a seminorm on l (Z). Furthermore, for every integer k 3, a U(k) a U(k ). This definition and result can be extended to the case k =. Property (P 0 ) means that the averages of the sequence a over the intervals I j converge to some it c(i, a). We have a U() = sup averages(a n ). For clarity, we explain what the definitions mean when k = 2. Property (P ) says that for every h Z, the averages (over n) on the intervals I i of a n+h a n converge to a it c h (I, a). Property (P 2 ) means that for all h, l Z, the averages (over n) on the intervals J j of a n+h+l a n+h a n+l a n converge to a it c (h,l) (J, a). We have that and cor 2 (I, a) = cor 2(J, a) = H + H + H H H 2 h=0 H h,l=0 c h (I, a) 2 c (h,l) (J, a). Remark. The seminorms U(k) are linked to the seminorms k of []. However there are important differences between the two different kind of seminorms. For example we have f 2k+ k+ = while a 2k+ U(k+) inf + f.t n f 2k k ā.σ n a 2k U(k) where σ denotes the shift; in general the inf is not a it and equality does not hold.

3 UIFORMITY SEMIORMS AD A IVERSE THEOREM 3 Proposition (Cauchy-Schwartz-Gowers Inequality). Let k 2 be an integer and for every ɛ {0, } k, let a(ɛ) = (a n (ɛ): n Z) be a bounded sequence. Let I = (I j : j ) be a sequence of intervals with I j + and assume that for every h = (h,..., h k ) Z k, the it ρ h := j + I j ɛ {0,} k C ɛ a n+ɛ h (ɛ) exists. Then the averages of ρ h (for h Z k ) converge and averages (ρ h ) a(ɛ) U(k). ɛ {0,} k 2. Inverse Theorem We recall the standard definition of a nilsystem and that of a nilsequence (see [2]): Definition 2. If G is a k-step nilpotent Lie group and Γ G is a discrete and cocompact subgroup, then the compact manifold X = G/Γ is called a k-step nilmanifold. The action of G on X by left translation is written (g, x) g x for g G and x X. A k-step nilmanifold X = G/Γ, endowed with the translation T : x τ x by some fixed element τ of G is called a k-step nilsystem. If f : X C is a continuous function, τ G and x 0 X, then (f(τ n x 0 ): n Z) is called a basic k-step nilsequence. Without loss we can restrict to the case that the nilsystem (X, T ) is minimal. A k-step nilsequence is defined to be a uniform it of basic k-step nilsequences. We are now ready to state the main theorem: Theorem 2 (Inverse Theorem). Let k 2 be an integer. For every bounded sequence a = (a n : n Z), the following properties are equivalent: (i) a U(k) = 0 ; (ii) For every (k )-step nilsequence b = (b n : n Z), the averages of the sequence (a n b n ) converge to 0. Remark 2. Assume that a bounded sequence a satisfies a k > 0. This means that there exists a sequence of intervals I = (I j : j ) such that the averages c h (I, a) are large for sufficiently many values of h. Theorem 2 says that there exist a (k )-step nilsequence b and a sequence of intervals J = (J l : l Z) such that the averages of the sequence (a n b n : n Z) on these intervals are large, but it says nothing about the relation between the intervals I j and J l, and in particular between their lengths. It is easy to build examples showing that Theorem 2 can not be improved in this direction and that this itation is intrinsic in this question. The implication (i) = (ii) of Theorem 2 follows from : Proposition 2. Let k 2 be an integer and let b = (b n : n Z) be a (k )-step nilsequence. For every δ > 0, there exists a constant C = C(b, δ) such that for every bounded sequence a = (a n : n Z) with a, sup averages(a n b n ) δ + C a U(k).

4 4 BERARD HOST AD BRYA KRA 3. Quantitative results: The dual norm Proposition 3 (and definition). Let (X, T, µ) be a minimal (k )-step nilsystem and let f be a smooth function on X. Then there exists a constant c such that f h dµ c h k for all h C(X), where C(X) denotes continuous functions on X. We set f k to be the smallest constant c satisfying this condition. The hypothesis of smoothness of f is too strong: the result still holds if f is Lipschitz (for some smooth metric on X). A similar result for the finite case appears in [3]. Proposition 4. Let (X, T, µ) be a minimal (k )-step nilsystem, x 0 X and f be a smooth function on X. Then for all bounded sequences a = (a n : n Z), we have sup averages(a n f(t n x 0 )) f k a U(k). Theorem 3 (Quantitative Inverse Theorem). Let a = (a n : n Z) be a bounded sequence with a U(k) > 0. Then for all δ > 0, there exists a k step nilsystem, x 0 X and a smooth function f on X such that f k = and sup averages(a n f(t n x 0 )) a U(k) δ. 4. The case k = 2 For k = 2, we have more explicit (and easier) results. We write T = R/Z and for t T, we write e(t) = exp(2πit). Recall that -step nilsequences are almost periodic sequences. Let f be a smooth function on a -step nilmanifold, that is, a compact abelian Lie group Z endowed with a minimal translation T. Then f can be written f(x) = f(χ)χ(x) with f(χ) < + χ Z b j= and we have f U(2) = ( f(χ) 4) /4 and f U(2) = ( f(χ) 4/3) 3/4. j= Proposition 2 for k = 2 can be deduced from: Lemma. For every bounded sequence a = (a n : n Z) and every sequence of intervals I = (I j : j Z) with I j +, we have sup sup a n e(nt) a U(2). j + t T I j Corollary. For k = 2, the equivalent properties (i) and (ii) of Theorem 2 are also equivalent to (iii) For every t T, sup averages(a n e(nt)) = 0. We adopt the convention of writing a measure preserving system as (X, µ, T ), omitting mention of the σ-algebra. j=

5 UIFORMITY SEMIORMS AD A IVERSE THEOREM 5 (iv) For every sequence I = (I j : j Z) of intervals with I j +, sup a n e(nt) = 0. j + t T I j For k = 3 a similar result holds, where the family of exponential sequences is replaced by a family of special nilsequences (this is the class M of [4]). 5. An application to ergodic theory Proposition 5. Let k 2 be an integer and let a be a bounded sequence. If a U(k) = 0, then for all measure preserving systems (X, µ, T ) and all functions f,..., f k L (µ), the averages a n T n f.t 2n f 2..T (k )n f k converge to zero in L 2 (µ) as +. The converse implication holds for k = 2. We believe that the converse implication holds also for k = 3, but the proof is not yet written (perhaps will be in the next few days); we conjecture that it holds for all k. References [] B. Host and B. Kra, onconventional ergodic averages and nilmanifolds. Annals of Math. 6, (2005) [2] V. Bergelson, B. Host and B. Kra, with an Appendix by I. Ruzsa. Multiple recurrence and nilsequences. Inventiones Math. 60 (2005), [3] B. Green and T. Tao. Linear equations in the primes. To appear, Annals of Math. [4] B. Host and B. Kra. Analysis of two step nilsequences. Preprint. Université Paris-Est, Laboratoire d analyse et de mathématiques appliquées, UMR CRS 8050, 5 bd Descartes, Marne la Vallée Cedex 2, France address: bernard.host@univ-mlv.fr Department of Mathematics, orthwestern University, 2033 Sheridan Road, Evanston, IL , USA address: kra@math.northwestern.edu