In this expository article, we describe the recent approach, motivated by ergodic theory, towards... more In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing in [26] that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to identify precisely what the obstructions could be that prevent the primes (or any other set) from behaving "randomly", and then either show that the obstructions do not actually occur, or else convert the obstructions into usable structural information on the primes.
We state and prove a quantitative inverse theorem for the Gowers uniformity norm U 3 (G) on an ar... more We state and prove a quantitative inverse theorem for the Gowers uniformity norm U 3 (G) on an arbitrary finite group G; the cases when G was of odd order or a vector space over F 2 had previously been established by Green and the second author and by Samorodnitsky respectively by Fourier-analytic methods, which we also employ here. We also prove a qualitative version of this inverse theorem using a structure theorem of Host-Kra type for ergodic Z ω-actions of order 2 on probability spaces established recently by Shalom and the authors.
Communications in Mathematical Physics, Aug 12, 2004
We show that in dimensions n ≥ 6 that one has global regularity for the Maxwell-Klein-Gordon equa... more We show that in dimensions n ≥ 6 that one has global regularity for the Maxwell-Klein-Gordon equations in the Coulomb gauge provided that the critical Sobolev normḢ n/2−1 ×Ḣ n/2−2 of the initial data is sufficiently small. These results are analogous to those recently obtained for the high-dimensional wave map equation [17], [7], [14], [12] but unlike the wave map equation, the Coulomb gauge non-linearity cannot be iterated away directly. We shall use a different approach, proving Strichartz estimates for the covariant wave equation. This in turn will be achieved by use of Littlewood-Paley multipliers, and a global parametrix for the covariant wave equation constructed using a truncated, microlocalized Cronstrom gauge. 1 This is of course not a very geometric operator, as we have omitted the terms involving A 0 and divA, as well as cubic terms of the form A 2 φ. However, the cubic terms and A 0 terms turn out to be negligible, and the divA term vanishes thanks to the Coulomb gauge condition (2). As it turns out, we will not use much covariant structure in our arguments, although the presence of the i in the lower-order term 2iA • ∇x is crucial in order for a certain phase correction to be purely imaginary. We should remark however that in three dimensions, recent work of Machedon and Sterbenz [11] has shown that the full covariant structure of the equation, including the "elliptic" component A 0 of the gauge, is essential to obtain optimal regularity results.
As usual, we use O(X) to denote a quantity bounded in magnitude by CX for some absolute constant ... more As usual, we use O(X) to denote a quantity bounded in magnitude by CX for some absolute constant X; if we need C to depend on various parameters, we will indicate this by subscripts. Thus for instance Oε,F (1) is a quantity bounded in magnitude by some expression Cε,F depending on ε, F. 2 We use here the expectation notation Ea∈Af (a) := 1 |A| a∈A f (a) for any finite nonempty set A, where |A| denotes the cardinality of A.
Let λ denote the Liouville function. We show that, as X → ∞, 2X X sup P (Y)∈R[Y ] deg P ≤k x≤n≤x+... more Let λ denote the Liouville function. We show that, as X → ∞, 2X X sup P (Y)∈R[Y ] deg P ≤k x≤n≤x+H λ(n)e(−P (n)) dx = o(XH) for all fixed k and X θ ≤ H ≤ X with 0 < θ < 1 fixed but arbitrarily small. Previously this was only established for k ≤ 1. We obtain this result as a special case of the corresponding statement for (non-pretentious) 1-bounded multiplicative functions that we prove. In fact, we are able to replace the polynomial phases e(−P (n)) by degree k nilsequences F (g(n)Γ). By the inverse theory for the Gowers norms this implies the higher order asymptotic uniformity result 2X X λ U k+1 ([x,x+H]) dx = o(X) in the same range of H. We present applications of this result to patterns of various types in the Liouville sequence. Firstly, we show that the number of sign patterns of the Liouville function is superpolynomial, making progress on a conjecture of Sarnak about the Liouville sequence having positive entropy. Secondly, we obtain cancellation in averages of λ over short polynomial progressions (n + P1(m),. .. , n + P k (m)), which in the case of linear polynomials yields a new averaged version of Chowla's conjecture. We are in fact able to prove our results on polynomial phases in the wider range H ≥ exp((log X) 5/8+ε), thus strengthening also previous work on the Fourier uniformity of the Liouville function.
Bilinear estimates for the wave equation in Minkowski space are normally proven using the Fourier... more Bilinear estimates for the wave equation in Minkowski space are normally proven using the Fourier transform and Plancherel's theorem. However, such methods are difficult to carry over to non-flat situations (such as wave equations with rough metrics, or connections with non-zero curvature). In this note we describe an alternative physical space approach which relies on vector fields, energy estimates as well as tube localization, splitting into coarse and fine scales, and induction on scales (in the spirit of Wolff [29], [30]). 1991 Mathematics Subject Classification. 35J10. 1 To avoid technicalities we shall assume that our waves are Schwartz functions in space. 2 Here and in the sequel, we use C to denote various constants depending only on n and a parameter ε which will appear later in the paper, and we use A B to denote the estimate A ≤ CB. Finally, we use A ∼ B to denote the estimate A B A. 3 The time independence property is invalid in a curved spacetime. There are however techniques, based on paradifferential calculus, which allow one to circumvent this difficulty, see [12] and [15].
Recently Wolff [25] obtained a nearly sharp L 2 bilinear restriction theorem for bounded subsets ... more Recently Wolff [25] obtained a nearly sharp L 2 bilinear restriction theorem for bounded subsets of the cone in general dimension. We obtain the endpoint of Wolff's estimate and generalize to the case when one of the subsets is large. As a consequence, we are able to deduce some nearly-sharp L p null form estimates.
Proceedings of the Edinburgh Mathematical Society, Feb 1, 2008
There has been much recent progress in the study of arithmetic progressions in various sets, such... more There has been much recent progress in the study of arithmetic progressions in various sets, such as dense subsets of the integers or of the primes. One key tool in these developments has been the sequence of Gowers uniformity norms U d (G), d = 1, 2, 3,. .. , on a finite additive group G; in particular, to detect arithmetic progressions of length k in G it is important to know under what circumstances the U k−1 (G) norm can be large. The U 1 (G) norm is trivial, and the U 2 (G) norm can be easily described in terms of the Fourier transform. In this paper we systematically study the U 3 (G) norm, defined for any function f : G → C on a finite additive group G by the formula f U 3 (G) := |G| −4 x,a,b,c∈G (f (x)f (x + a)f (x + b)f (x + c)f (x + a + b) × f (x + b + c)f (x + c + a)f (x + a + b + c)) 1/8. We give an inverse theorem for the U 3 (G) norm on an arbitrary group G. In the finite-field case G = F n 5 we show that a bounded function f : G → C has large U 3 (G) norm if and only if it has a large inner product with a function e(φ), where e(x) := e 2πix and φ : F n 5 → R/Z is a quadratic phase function. In a general G the statement is more complicated: the phase φ is quadratic only locally on a Bohr neighbourhood in G. As an application we extend Gowers's proof of Szemerédi's theorem for progressions of length four to arbitrary abelian G. More precisely, writing r 4 (G) for the size of the largest A ⊆ G which does not contain a progression of length four, we prove that r 4 (G) |G|(log log |G|) −c , where c is an absolute constant. We also discuss links between our ideas and recent results of Host, Kra and Ziegler in ergodic theory. In future papers we will apply variants of our inverse theorems to obtain an asymptotic for the number of quadruples p 1 < p 2 < p 3 < p 4 N of primes in arithmetic progression, and to obtain significantly stronger bounds for r 4 (G).
Let P 1 ,. .. , P k : Z → Z be polynomials of degree at most d for some d 1, with the degree d co... more Let P 1 ,. .. , P k : Z → Z be polynomials of degree at most d for some d 1, with the degree d coefficients all distinct, and admissible in the sense that for every prime p, there exists integers n, m such that n + P 1 (m),. .. , n + P k (m) are all not divisible by p. We show that there exist infinitely many natural numbers n, m such that n + P 1 (m),. .. , n + P k (m) are simultaneously prime, generalizing a previous result of the authors, which was restricted to the special case P 1 (0) = • • • = P k (0) = 0 (though it allowed for the top degree coefficients to coincide). Furthermore, we obtain an asymptotic for the number of such prime pairs n, m with n N and m M with M slightly less than N 1/d. This asymptotic is already new in general in the homogeneous case P 1 (0) = • • • = P k (0) = 0. Our arguments rely on four ingredients. The first is a (slightly modified) generalized von Neumann theorem of the authors, reducing matters to controlling certain averaged local Gowers norms of (suitable normalizations of) the von Mangoldt function. The second is a more recent concatenation theorem of the authors, controlling these averaged local Gowers norms by global Gowers norms. The third ingredient is the work of Green and the authors on linear equations in primes, allowing one to compute these global Gowers norms for the normalized von Mangoldt functions. Finally, we use the Conlon-Fox-Zhao densification approach to the transference principle to combine the preceding three ingredients together. In the special case P 1 (0) = • • • = P k (0) = 0, our methods also give infinitely many n, m with n + P 1 (m),. .. , n + P k (m) in a specified set primes of positive relative density δ, with m bounded by log L n for some L independent of the density δ. This improves slightly on a result from our previous paper, in which L was allowed to depend on δ.
Mathematical proceedings of the Cambridge Philosophical Society, Mar 24, 2010
We establish a correspondence between inverse sumset theorems (which can be viewed as classificat... more We establish a correspondence between inverse sumset theorems (which can be viewed as classifications of approximate (abelian) groups) and inverse theorems for the Gowers norms (which can be viewed as classifications of approximate polynomials). In particular, we show that the inverse sumset theorems of Freȋman type are equivalent to the known inverse results for the Gowers U 3 norms, and moreover that the conjectured polynomial strengthening of the former is also equivalent to the polynomial strengthening of the latter. We establish this equivalence in two model settings, namely that of the finite field vector spaces F n 2 , and of the cyclic groups Z/N Z. In both cases the argument involves clarifying the structure of certain types of approximate homomorphism.
This paper is a part of our programme to generalise the Hardy-Littlewood method to handle systems... more This paper is a part of our programme to generalise the Hardy-Littlewood method to handle systems of linear questions in primes. This programme is laid out in our paper Linear equations in primes [14]. In particular, the results of this paper may be used, together with the machinery of [14], to establish an asymptotic for the number of four-term progressions p 1 < p 2 < p 3 < p 4 N of primes, and more generally any problem counting prime points inside a "non-degenerate" affine lattice of codimension at most 2. The main result of this paper is a proof of the Möbius and Nilsequences Conjecture for 1 and 2-step nilsequences. This conjecture is introduced in [14] and amounts to showing that if G/Γ is an s-step nilmanifold, s 2, if F : G/Γ → [−1, 1] is a Lipschitz function, and if T g : G/Γ → G/Γ is the action of g ∈ G on G/Γ, then N −1 n N µ(n)F (T n g • x) ≪ A,G/Γ F Lip log −A N uniformly in g ∈ G and x ∈ G/Γ, for any A > 0. This can be viewed as a "quadratic" generalisation of an exponential sum estimate of Davenport [7], and is proven by following the methods of Vinogradov and Vaughan.
As usual, we use O(X) to denote a quantity bounded in magnitude by CX for some absolute constant ... more As usual, we use O(X) to denote a quantity bounded in magnitude by CX for some absolute constant X; if we need C to depend on various parameters, we will indicate this by subscripts. Thus for instance Oε,F (1) is a quantity bounded in magnitude by some expression Cε,F depending on ε, F. 2 We use here the expectation notation Ea∈Af (a) := 1 |A| a∈A f (a) for any finite nonempty set A, where |A| denotes the cardinality of A.
We prove the inverse conjecture for the Gowers U s+1 [N ]-norm for all s 1; this is new for s 4. ... more We prove the inverse conjecture for the Gowers U s+1 [N ]-norm for all s 1; this is new for s 4. More precisely, we establish that if f : [N ] → [−1, 1] is a function with f U s+1 [N ] δ, then there is a boundedcomplexity s-step nilsequence F (g(n)Γ) that correlates with f , where the bounds on the complexity and correlation depend only on s and δ. From previous results, this conjecture implies the Hardy-Littlewood prime tuples conjecture for any linear system of finite complexity. Contents 1. Introduction 2. Strategy of the proof 3. Basic notation 4. The polynomial formulation of GI(s) 5. Taking ultralimits 6. Nilcharacters and symbols in one and several variables 7. A more detailed outline of the argument 8. A variant of Gowers's Cauchy-Schwarz argument 9. Frequencies and representations 10. Linear independence and the sunflower lemma 11. Obtaining bracket-linear behaviour 12. Building a nilobject 13. The symmetry argument Appendix A. Basic theory of ultralimits Appendix B. Polynomial algebra
A theorem of Leibman asserts that a polynomial orbit (g(n)Γ) n∈Z on a nilmanifold G/Γ is always e... more A theorem of Leibman asserts that a polynomial orbit (g(n)Γ) n∈Z on a nilmanifold G/Γ is always equidistributed in a union of closed subnilmanifolds of G/Γ. In this paper we give a quantitative version of Leibman's result, describing the uniform distribution properties of a finite polynomial orbit (g(n)Γ) n∈[N ] in a nilmanifold. More specifically we show that there is a factorisation g = εg γ, where ε(n) is "smooth," (γ(n)Γ) n∈Z is periodic and "rational," and (g (n)Γ)n∈P is uniformly distributed (up to a specified error δ) inside some subnilmanifold G /Γ of G/Γ for all sufficiently dense arithmetic progressions P ⊆ [N ]. Our bounds are uniform in N and are polynomial in the error tolerance δ. In a companion paper we shall use this theorem to establish the Möbius and Nilsequences conjecture from an earlier paper of ours.
We study higher uniformity properties of the Möbius function µ, the von Mangoldt function Λ, and ... more We study higher uniformity properties of the Möbius function µ, the von Mangoldt function Λ, and the divisor functions d k on short intervals (X, X + H] with X θ+ε ≤ H ≤ X 1−ε for a fixed constant 0 ≤ θ < 1 and any ε > 0. More precisely, letting Λ ♯ and d ♯ k be suitable approximants of Λ and d k and µ ♯ = 0, we show for instance that, for any nilsequence F (g(n)Γ), we have X<n≤X+H (f (n) − f ♯ (n))F (g(n)Γ) ≪ H log −A X when θ = 5/8 and f ∈ {Λ, µ, d k } or θ = 1/3 and f = d2. As a consequence, we show that the short interval Gowers norms f − f ♯ U s (X,X+H] are also asymptotically small for any fixed s for these choices of f, θ. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals, and show that multiple ergodic averages along primes in short intervals converge in L 2. Our innovations include the use of multi-parameter nilsequence equidistribution theorems to control type II sums, and an elementary decomposition of the neighbourhood of a hyperbola into arithmetic progressions to control type I2 sums. 2 K. Matomäki, X. Shao, T. Tao, and J. Teräväinen 5. The type I case 56 6. The non-abelian type II case 57 7. The abelian type II case 63 8. The type I 2 case 67 9. Controlling the Gowers uniformity norms 75 10. Applications 85 References 88 * ≪ ε,d,D δ −O d,D (1) H log 3 4 k−1 X. 1 For definitions of undefined terms such as "filtered nilmanifold" and Poly(Z → G), see Definitions 2.6 and 2.5 below. For our conventions for asymptotic notation such as ≪, see Section 1.5.
We revisit the multilinear Kakeya, curved Kakeya, restriction, and oscillatory integral estimates... more We revisit the multilinear Kakeya, curved Kakeya, restriction, and oscillatory integral estimates that were obtained in a paper of Bennett, Carbery, and the author using a heat flow monotonicity method applied to a fractional Cartesian product, together with induction on scales arguments. Many of these estimates contained losses of the form R ε (or log O(1) R) for some scale factor R. By further developing the heat flow method, and applying it directly for the first time to the multilinear curved Kakeya and restriction settings, we are able to eliminate these losses, as long as the exponent p stays away from the endpoint. In particular, we establish global multilinear restriction estimates away from the endpoint, without any curvature hypotheses on the hypersurfaces.
We demonstrate the (H 1 , L 1,2) or (L p , L p,2) mapping properties of several rough operators. ... more We demonstrate the (H 1 , L 1,2) or (L p , L p,2) mapping properties of several rough operators. In all cases these estimates are sharp in the sense that the Lorentz exponent 2 cannot be replaced by any lower number. 1991 Mathematics Subject Classification. 42B20.
In a recent paper [18], Kenig, Ponce and Vega study the low regularity behavior of the focusing n... more In a recent paper [18], Kenig, Ponce and Vega study the low regularity behavior of the focusing nonlinear Schrödinger (NLS), focusing modified Korteweg-de Vries (mKdV), and complex Korteweg-de Vries (KdV) equations. Using soliton and breather solutions, they demonstrate the lack of local well-posedness for these equations below their respective endpoint regularities. In this paper, we study the defocusing analogues of these equations, namely defocusing NLS, defocusing mKdV, and real KdV, all in one spatial dimension, for which suitable soliton and breather solutions are unavailable. We construct for each of these equations classes of modified scattering solutions, which exist globally in time, and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts. These solutions are used to demonstrate lack of local well-posedness in certain Sobolev spaces, in the sense that the dependence of solutions upon initial data fails to be uniformly continuous. In particular, we show that the mKdV flow is not uniformly continuous in the L 2 topology, despite the existence of global weak solutions at this regularity. Finally, we investigate the KdV equation at the endpoint regularity H −3/4 , and construct solutions for both the real and complex KdV equations. The construction provides a nontrivial time interval [−T, T ] and a locally Lipschitz continuous map taking the initial data in H −3/4 to a distributional solution u ∈ C 0 ([−T, T ]; H −3/4) which is uniquely defined for all smooth data. The proof uses a generalized Miura transform to transfer the existing endpoint regularity theory for mKdV to KdV. Contents 1. Introduction 1.1. The defocusing nonlinear Schrödinger equation 1.2. The defocusing modified Korteweg-de Vries equation 1.3. The Korteweg-de Vries equation 1.4. Periodic analogues 2. Notation and modulation bounds 3. The pseudo-conformal transformation 3.1. Definition of the pc transform and some basic properties 3.2. Relation to Schrödinger equations 3.3. Decoherence and lack of scattering in L 2 4. Ill-posedness of NLS 4.1. The subcritical case 4.2. The critical case
Let K 1 be a parameter. A K-approximate group is a finite set A in a (local) group which contains... more Let K 1 be a parameter. A K-approximate group is a finite set A in a (local) group which contains the identity, is symmetric, and such that A • A is covered by K left translates of A. The main result of this paper is a qualitative description of approximate groups as being essentially finite-by-nilpotent, answering a conjecture of H. Helfgott and E. Lindenstrauss. This may be viewed as a generalisation of the Freiman-Ruzsa theorem on sets of small doubling in the integers to arbitrary groups. We begin by establishing a correspondence principle between approximate groups and locally compact (local) groups that allows us to recover many results recently established in a fundamental paper of Hrushovski. In particular we establish that approximate groups can be approximately modeled by Lie groups. To prove our main theorem we apply some additional arguments essentially due to Gleason. These arose in the solution of Hilbert's fifth problem in the 1950s. Applications of our main theorem include a finitary refinement of Gromov's theorem, as well as a generalized Margulis lemma conjectured by Gromov and a result on the virtual nilpotence of the fundamental group of Ricci almost nonnegatively curved manifolds. Résumé. Soit K 1 un paramètre. On appelle groupe K-approximatif toute partie finie A d'un groupe (ou d'un groupe local) qui est symétrique, contient l'identité et est telle que A • A peutêtre recouvert par au plus K translatésà gauche de A. Le résultat principal de cet article montre que tout groupe approximatif est, grossièrement, finipar-nilpotent, ce qui répond par l'affirmativeà une conjecture de H.Helfgott et de E.Lindenstrauss. On peut interpréter ce théorème comme une généralisationà un groupe quelconque du théorème de Freiman-Ruzsa sur la structure des parties finiesà petit doublement du groupe additif des entiers relatifs. Nous commençons parétablir un principe de correspondence entre les groupes approximatifs et les groupes (ou groupes locaux) localement compacts, puis nous en déduisons de nombreuses conséquences issues d'un important article récent de Hrushovski. En particulier, nous montrons que tout groupe approximatif peut-être représenté par un groupe de Lie. Pour démontrer notre théorème principal, nous appliquons des arguments, en substance dusà Gleason pour la plupart, qui ont vus le jour dans le contexte de la solution du cinquième problème de Hilbert dans les années 50. En guise d'application du théorème principal, nous montrons une version fine du théorème de Gromov, ainsi qu'un lemme de Margulis généralisé conjecturé par Gromov, et un résultat sur la presque nilpotence des groupes fondamentaux des variétésà courbure de Ricci presque positive.
In this expository article, we describe the recent approach, motivated by ergodic theory, towards... more In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing in [26] that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to identify precisely what the obstructions could be that prevent the primes (or any other set) from behaving "randomly", and then either show that the obstructions do not actually occur, or else convert the obstructions into usable structural information on the primes.
We state and prove a quantitative inverse theorem for the Gowers uniformity norm U 3 (G) on an ar... more We state and prove a quantitative inverse theorem for the Gowers uniformity norm U 3 (G) on an arbitrary finite group G; the cases when G was of odd order or a vector space over F 2 had previously been established by Green and the second author and by Samorodnitsky respectively by Fourier-analytic methods, which we also employ here. We also prove a qualitative version of this inverse theorem using a structure theorem of Host-Kra type for ergodic Z ω-actions of order 2 on probability spaces established recently by Shalom and the authors.
Communications in Mathematical Physics, Aug 12, 2004
We show that in dimensions n ≥ 6 that one has global regularity for the Maxwell-Klein-Gordon equa... more We show that in dimensions n ≥ 6 that one has global regularity for the Maxwell-Klein-Gordon equations in the Coulomb gauge provided that the critical Sobolev normḢ n/2−1 ×Ḣ n/2−2 of the initial data is sufficiently small. These results are analogous to those recently obtained for the high-dimensional wave map equation [17], [7], [14], [12] but unlike the wave map equation, the Coulomb gauge non-linearity cannot be iterated away directly. We shall use a different approach, proving Strichartz estimates for the covariant wave equation. This in turn will be achieved by use of Littlewood-Paley multipliers, and a global parametrix for the covariant wave equation constructed using a truncated, microlocalized Cronstrom gauge. 1 This is of course not a very geometric operator, as we have omitted the terms involving A 0 and divA, as well as cubic terms of the form A 2 φ. However, the cubic terms and A 0 terms turn out to be negligible, and the divA term vanishes thanks to the Coulomb gauge condition (2). As it turns out, we will not use much covariant structure in our arguments, although the presence of the i in the lower-order term 2iA • ∇x is crucial in order for a certain phase correction to be purely imaginary. We should remark however that in three dimensions, recent work of Machedon and Sterbenz [11] has shown that the full covariant structure of the equation, including the "elliptic" component A 0 of the gauge, is essential to obtain optimal regularity results.
As usual, we use O(X) to denote a quantity bounded in magnitude by CX for some absolute constant ... more As usual, we use O(X) to denote a quantity bounded in magnitude by CX for some absolute constant X; if we need C to depend on various parameters, we will indicate this by subscripts. Thus for instance Oε,F (1) is a quantity bounded in magnitude by some expression Cε,F depending on ε, F. 2 We use here the expectation notation Ea∈Af (a) := 1 |A| a∈A f (a) for any finite nonempty set A, where |A| denotes the cardinality of A.
Let λ denote the Liouville function. We show that, as X → ∞, 2X X sup P (Y)∈R[Y ] deg P ≤k x≤n≤x+... more Let λ denote the Liouville function. We show that, as X → ∞, 2X X sup P (Y)∈R[Y ] deg P ≤k x≤n≤x+H λ(n)e(−P (n)) dx = o(XH) for all fixed k and X θ ≤ H ≤ X with 0 < θ < 1 fixed but arbitrarily small. Previously this was only established for k ≤ 1. We obtain this result as a special case of the corresponding statement for (non-pretentious) 1-bounded multiplicative functions that we prove. In fact, we are able to replace the polynomial phases e(−P (n)) by degree k nilsequences F (g(n)Γ). By the inverse theory for the Gowers norms this implies the higher order asymptotic uniformity result 2X X λ U k+1 ([x,x+H]) dx = o(X) in the same range of H. We present applications of this result to patterns of various types in the Liouville sequence. Firstly, we show that the number of sign patterns of the Liouville function is superpolynomial, making progress on a conjecture of Sarnak about the Liouville sequence having positive entropy. Secondly, we obtain cancellation in averages of λ over short polynomial progressions (n + P1(m),. .. , n + P k (m)), which in the case of linear polynomials yields a new averaged version of Chowla's conjecture. We are in fact able to prove our results on polynomial phases in the wider range H ≥ exp((log X) 5/8+ε), thus strengthening also previous work on the Fourier uniformity of the Liouville function.
Bilinear estimates for the wave equation in Minkowski space are normally proven using the Fourier... more Bilinear estimates for the wave equation in Minkowski space are normally proven using the Fourier transform and Plancherel's theorem. However, such methods are difficult to carry over to non-flat situations (such as wave equations with rough metrics, or connections with non-zero curvature). In this note we describe an alternative physical space approach which relies on vector fields, energy estimates as well as tube localization, splitting into coarse and fine scales, and induction on scales (in the spirit of Wolff [29], [30]). 1991 Mathematics Subject Classification. 35J10. 1 To avoid technicalities we shall assume that our waves are Schwartz functions in space. 2 Here and in the sequel, we use C to denote various constants depending only on n and a parameter ε which will appear later in the paper, and we use A B to denote the estimate A ≤ CB. Finally, we use A ∼ B to denote the estimate A B A. 3 The time independence property is invalid in a curved spacetime. There are however techniques, based on paradifferential calculus, which allow one to circumvent this difficulty, see [12] and [15].
Recently Wolff [25] obtained a nearly sharp L 2 bilinear restriction theorem for bounded subsets ... more Recently Wolff [25] obtained a nearly sharp L 2 bilinear restriction theorem for bounded subsets of the cone in general dimension. We obtain the endpoint of Wolff's estimate and generalize to the case when one of the subsets is large. As a consequence, we are able to deduce some nearly-sharp L p null form estimates.
Proceedings of the Edinburgh Mathematical Society, Feb 1, 2008
There has been much recent progress in the study of arithmetic progressions in various sets, such... more There has been much recent progress in the study of arithmetic progressions in various sets, such as dense subsets of the integers or of the primes. One key tool in these developments has been the sequence of Gowers uniformity norms U d (G), d = 1, 2, 3,. .. , on a finite additive group G; in particular, to detect arithmetic progressions of length k in G it is important to know under what circumstances the U k−1 (G) norm can be large. The U 1 (G) norm is trivial, and the U 2 (G) norm can be easily described in terms of the Fourier transform. In this paper we systematically study the U 3 (G) norm, defined for any function f : G → C on a finite additive group G by the formula f U 3 (G) := |G| −4 x,a,b,c∈G (f (x)f (x + a)f (x + b)f (x + c)f (x + a + b) × f (x + b + c)f (x + c + a)f (x + a + b + c)) 1/8. We give an inverse theorem for the U 3 (G) norm on an arbitrary group G. In the finite-field case G = F n 5 we show that a bounded function f : G → C has large U 3 (G) norm if and only if it has a large inner product with a function e(φ), where e(x) := e 2πix and φ : F n 5 → R/Z is a quadratic phase function. In a general G the statement is more complicated: the phase φ is quadratic only locally on a Bohr neighbourhood in G. As an application we extend Gowers's proof of Szemerédi's theorem for progressions of length four to arbitrary abelian G. More precisely, writing r 4 (G) for the size of the largest A ⊆ G which does not contain a progression of length four, we prove that r 4 (G) |G|(log log |G|) −c , where c is an absolute constant. We also discuss links between our ideas and recent results of Host, Kra and Ziegler in ergodic theory. In future papers we will apply variants of our inverse theorems to obtain an asymptotic for the number of quadruples p 1 < p 2 < p 3 < p 4 N of primes in arithmetic progression, and to obtain significantly stronger bounds for r 4 (G).
Let P 1 ,. .. , P k : Z → Z be polynomials of degree at most d for some d 1, with the degree d co... more Let P 1 ,. .. , P k : Z → Z be polynomials of degree at most d for some d 1, with the degree d coefficients all distinct, and admissible in the sense that for every prime p, there exists integers n, m such that n + P 1 (m),. .. , n + P k (m) are all not divisible by p. We show that there exist infinitely many natural numbers n, m such that n + P 1 (m),. .. , n + P k (m) are simultaneously prime, generalizing a previous result of the authors, which was restricted to the special case P 1 (0) = • • • = P k (0) = 0 (though it allowed for the top degree coefficients to coincide). Furthermore, we obtain an asymptotic for the number of such prime pairs n, m with n N and m M with M slightly less than N 1/d. This asymptotic is already new in general in the homogeneous case P 1 (0) = • • • = P k (0) = 0. Our arguments rely on four ingredients. The first is a (slightly modified) generalized von Neumann theorem of the authors, reducing matters to controlling certain averaged local Gowers norms of (suitable normalizations of) the von Mangoldt function. The second is a more recent concatenation theorem of the authors, controlling these averaged local Gowers norms by global Gowers norms. The third ingredient is the work of Green and the authors on linear equations in primes, allowing one to compute these global Gowers norms for the normalized von Mangoldt functions. Finally, we use the Conlon-Fox-Zhao densification approach to the transference principle to combine the preceding three ingredients together. In the special case P 1 (0) = • • • = P k (0) = 0, our methods also give infinitely many n, m with n + P 1 (m),. .. , n + P k (m) in a specified set primes of positive relative density δ, with m bounded by log L n for some L independent of the density δ. This improves slightly on a result from our previous paper, in which L was allowed to depend on δ.
Mathematical proceedings of the Cambridge Philosophical Society, Mar 24, 2010
We establish a correspondence between inverse sumset theorems (which can be viewed as classificat... more We establish a correspondence between inverse sumset theorems (which can be viewed as classifications of approximate (abelian) groups) and inverse theorems for the Gowers norms (which can be viewed as classifications of approximate polynomials). In particular, we show that the inverse sumset theorems of Freȋman type are equivalent to the known inverse results for the Gowers U 3 norms, and moreover that the conjectured polynomial strengthening of the former is also equivalent to the polynomial strengthening of the latter. We establish this equivalence in two model settings, namely that of the finite field vector spaces F n 2 , and of the cyclic groups Z/N Z. In both cases the argument involves clarifying the structure of certain types of approximate homomorphism.
This paper is a part of our programme to generalise the Hardy-Littlewood method to handle systems... more This paper is a part of our programme to generalise the Hardy-Littlewood method to handle systems of linear questions in primes. This programme is laid out in our paper Linear equations in primes [14]. In particular, the results of this paper may be used, together with the machinery of [14], to establish an asymptotic for the number of four-term progressions p 1 < p 2 < p 3 < p 4 N of primes, and more generally any problem counting prime points inside a "non-degenerate" affine lattice of codimension at most 2. The main result of this paper is a proof of the Möbius and Nilsequences Conjecture for 1 and 2-step nilsequences. This conjecture is introduced in [14] and amounts to showing that if G/Γ is an s-step nilmanifold, s 2, if F : G/Γ → [−1, 1] is a Lipschitz function, and if T g : G/Γ → G/Γ is the action of g ∈ G on G/Γ, then N −1 n N µ(n)F (T n g • x) ≪ A,G/Γ F Lip log −A N uniformly in g ∈ G and x ∈ G/Γ, for any A > 0. This can be viewed as a "quadratic" generalisation of an exponential sum estimate of Davenport [7], and is proven by following the methods of Vinogradov and Vaughan.
As usual, we use O(X) to denote a quantity bounded in magnitude by CX for some absolute constant ... more As usual, we use O(X) to denote a quantity bounded in magnitude by CX for some absolute constant X; if we need C to depend on various parameters, we will indicate this by subscripts. Thus for instance Oε,F (1) is a quantity bounded in magnitude by some expression Cε,F depending on ε, F. 2 We use here the expectation notation Ea∈Af (a) := 1 |A| a∈A f (a) for any finite nonempty set A, where |A| denotes the cardinality of A.
We prove the inverse conjecture for the Gowers U s+1 [N ]-norm for all s 1; this is new for s 4. ... more We prove the inverse conjecture for the Gowers U s+1 [N ]-norm for all s 1; this is new for s 4. More precisely, we establish that if f : [N ] → [−1, 1] is a function with f U s+1 [N ] δ, then there is a boundedcomplexity s-step nilsequence F (g(n)Γ) that correlates with f , where the bounds on the complexity and correlation depend only on s and δ. From previous results, this conjecture implies the Hardy-Littlewood prime tuples conjecture for any linear system of finite complexity. Contents 1. Introduction 2. Strategy of the proof 3. Basic notation 4. The polynomial formulation of GI(s) 5. Taking ultralimits 6. Nilcharacters and symbols in one and several variables 7. A more detailed outline of the argument 8. A variant of Gowers's Cauchy-Schwarz argument 9. Frequencies and representations 10. Linear independence and the sunflower lemma 11. Obtaining bracket-linear behaviour 12. Building a nilobject 13. The symmetry argument Appendix A. Basic theory of ultralimits Appendix B. Polynomial algebra
A theorem of Leibman asserts that a polynomial orbit (g(n)Γ) n∈Z on a nilmanifold G/Γ is always e... more A theorem of Leibman asserts that a polynomial orbit (g(n)Γ) n∈Z on a nilmanifold G/Γ is always equidistributed in a union of closed subnilmanifolds of G/Γ. In this paper we give a quantitative version of Leibman's result, describing the uniform distribution properties of a finite polynomial orbit (g(n)Γ) n∈[N ] in a nilmanifold. More specifically we show that there is a factorisation g = εg γ, where ε(n) is "smooth," (γ(n)Γ) n∈Z is periodic and "rational," and (g (n)Γ)n∈P is uniformly distributed (up to a specified error δ) inside some subnilmanifold G /Γ of G/Γ for all sufficiently dense arithmetic progressions P ⊆ [N ]. Our bounds are uniform in N and are polynomial in the error tolerance δ. In a companion paper we shall use this theorem to establish the Möbius and Nilsequences conjecture from an earlier paper of ours.
We study higher uniformity properties of the Möbius function µ, the von Mangoldt function Λ, and ... more We study higher uniformity properties of the Möbius function µ, the von Mangoldt function Λ, and the divisor functions d k on short intervals (X, X + H] with X θ+ε ≤ H ≤ X 1−ε for a fixed constant 0 ≤ θ < 1 and any ε > 0. More precisely, letting Λ ♯ and d ♯ k be suitable approximants of Λ and d k and µ ♯ = 0, we show for instance that, for any nilsequence F (g(n)Γ), we have X<n≤X+H (f (n) − f ♯ (n))F (g(n)Γ) ≪ H log −A X when θ = 5/8 and f ∈ {Λ, µ, d k } or θ = 1/3 and f = d2. As a consequence, we show that the short interval Gowers norms f − f ♯ U s (X,X+H] are also asymptotically small for any fixed s for these choices of f, θ. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals, and show that multiple ergodic averages along primes in short intervals converge in L 2. Our innovations include the use of multi-parameter nilsequence equidistribution theorems to control type II sums, and an elementary decomposition of the neighbourhood of a hyperbola into arithmetic progressions to control type I2 sums. 2 K. Matomäki, X. Shao, T. Tao, and J. Teräväinen 5. The type I case 56 6. The non-abelian type II case 57 7. The abelian type II case 63 8. The type I 2 case 67 9. Controlling the Gowers uniformity norms 75 10. Applications 85 References 88 * ≪ ε,d,D δ −O d,D (1) H log 3 4 k−1 X. 1 For definitions of undefined terms such as "filtered nilmanifold" and Poly(Z → G), see Definitions 2.6 and 2.5 below. For our conventions for asymptotic notation such as ≪, see Section 1.5.
We revisit the multilinear Kakeya, curved Kakeya, restriction, and oscillatory integral estimates... more We revisit the multilinear Kakeya, curved Kakeya, restriction, and oscillatory integral estimates that were obtained in a paper of Bennett, Carbery, and the author using a heat flow monotonicity method applied to a fractional Cartesian product, together with induction on scales arguments. Many of these estimates contained losses of the form R ε (or log O(1) R) for some scale factor R. By further developing the heat flow method, and applying it directly for the first time to the multilinear curved Kakeya and restriction settings, we are able to eliminate these losses, as long as the exponent p stays away from the endpoint. In particular, we establish global multilinear restriction estimates away from the endpoint, without any curvature hypotheses on the hypersurfaces.
We demonstrate the (H 1 , L 1,2) or (L p , L p,2) mapping properties of several rough operators. ... more We demonstrate the (H 1 , L 1,2) or (L p , L p,2) mapping properties of several rough operators. In all cases these estimates are sharp in the sense that the Lorentz exponent 2 cannot be replaced by any lower number. 1991 Mathematics Subject Classification. 42B20.
In a recent paper [18], Kenig, Ponce and Vega study the low regularity behavior of the focusing n... more In a recent paper [18], Kenig, Ponce and Vega study the low regularity behavior of the focusing nonlinear Schrödinger (NLS), focusing modified Korteweg-de Vries (mKdV), and complex Korteweg-de Vries (KdV) equations. Using soliton and breather solutions, they demonstrate the lack of local well-posedness for these equations below their respective endpoint regularities. In this paper, we study the defocusing analogues of these equations, namely defocusing NLS, defocusing mKdV, and real KdV, all in one spatial dimension, for which suitable soliton and breather solutions are unavailable. We construct for each of these equations classes of modified scattering solutions, which exist globally in time, and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts. These solutions are used to demonstrate lack of local well-posedness in certain Sobolev spaces, in the sense that the dependence of solutions upon initial data fails to be uniformly continuous. In particular, we show that the mKdV flow is not uniformly continuous in the L 2 topology, despite the existence of global weak solutions at this regularity. Finally, we investigate the KdV equation at the endpoint regularity H −3/4 , and construct solutions for both the real and complex KdV equations. The construction provides a nontrivial time interval [−T, T ] and a locally Lipschitz continuous map taking the initial data in H −3/4 to a distributional solution u ∈ C 0 ([−T, T ]; H −3/4) which is uniquely defined for all smooth data. The proof uses a generalized Miura transform to transfer the existing endpoint regularity theory for mKdV to KdV. Contents 1. Introduction 1.1. The defocusing nonlinear Schrödinger equation 1.2. The defocusing modified Korteweg-de Vries equation 1.3. The Korteweg-de Vries equation 1.4. Periodic analogues 2. Notation and modulation bounds 3. The pseudo-conformal transformation 3.1. Definition of the pc transform and some basic properties 3.2. Relation to Schrödinger equations 3.3. Decoherence and lack of scattering in L 2 4. Ill-posedness of NLS 4.1. The subcritical case 4.2. The critical case
Let K 1 be a parameter. A K-approximate group is a finite set A in a (local) group which contains... more Let K 1 be a parameter. A K-approximate group is a finite set A in a (local) group which contains the identity, is symmetric, and such that A • A is covered by K left translates of A. The main result of this paper is a qualitative description of approximate groups as being essentially finite-by-nilpotent, answering a conjecture of H. Helfgott and E. Lindenstrauss. This may be viewed as a generalisation of the Freiman-Ruzsa theorem on sets of small doubling in the integers to arbitrary groups. We begin by establishing a correspondence principle between approximate groups and locally compact (local) groups that allows us to recover many results recently established in a fundamental paper of Hrushovski. In particular we establish that approximate groups can be approximately modeled by Lie groups. To prove our main theorem we apply some additional arguments essentially due to Gleason. These arose in the solution of Hilbert's fifth problem in the 1950s. Applications of our main theorem include a finitary refinement of Gromov's theorem, as well as a generalized Margulis lemma conjectured by Gromov and a result on the virtual nilpotence of the fundamental group of Ricci almost nonnegatively curved manifolds. Résumé. Soit K 1 un paramètre. On appelle groupe K-approximatif toute partie finie A d'un groupe (ou d'un groupe local) qui est symétrique, contient l'identité et est telle que A • A peutêtre recouvert par au plus K translatésà gauche de A. Le résultat principal de cet article montre que tout groupe approximatif est, grossièrement, finipar-nilpotent, ce qui répond par l'affirmativeà une conjecture de H.Helfgott et de E.Lindenstrauss. On peut interpréter ce théorème comme une généralisationà un groupe quelconque du théorème de Freiman-Ruzsa sur la structure des parties finiesà petit doublement du groupe additif des entiers relatifs. Nous commençons parétablir un principe de correspondence entre les groupes approximatifs et les groupes (ou groupes locaux) localement compacts, puis nous en déduisons de nombreuses conséquences issues d'un important article récent de Hrushovski. En particulier, nous montrons que tout groupe approximatif peut-être représenté par un groupe de Lie. Pour démontrer notre théorème principal, nous appliquons des arguments, en substance dusà Gleason pour la plupart, qui ont vus le jour dans le contexte de la solution du cinquième problème de Hilbert dans les années 50. En guise d'application du théorème principal, nous montrons une version fine du théorème de Gromov, ainsi qu'un lemme de Margulis généralisé conjecturé par Gromov, et un résultat sur la presque nilpotence des groupes fondamentaux des variétésà courbure de Ricci presque positive.
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