Higher order fourier analysis

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Higher order fourier analysis

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Higher order Fourier analysis Terence Tao Department of Mathematics, UCLA, Los Angeles, CA 90095 E-mail address: tao@math.ucla.edu To Garth Gaudry, who set me on the road; To my family, for their constant support; And to the readers of my blog, for their feedback and contributions. Contents Preface ix Acknowledgments x Chapter 1. Higher order Fourier analysis 1 §1.1. Equidistribution of polynomial sequences in tori 2 §1.2. Roth’s theorem 31 §1.3. Linear patterns 54 §1.4. Equidistribution of polynomials over finite fields 71 §1.5. The inverse conjecture for the Gowers norm I. The finite field case 89 §1.6. The inverse conjecture for the Gowers norm II. The integer case 110 §1.7. Linear equations in primes 131 Chapter 2. Related articles 155 §2.1. Ultralimit analysis and quantitative algebraic geometry156 §2.2. Higher order Hilbert spaces 180 §2.3. The uncertainty principle 195 Bibliography 215 Index 221 vii Preface Traditionally, Fourier analysis has been focused the analysis of func- tions in terms of linear phase functions such as the sequence n → e(αn) = e 2πiαn . In recent years, though, applications have arisen - particularly in connection with problems involving linear patterns such as arithmetic progressions - in which it has been necessary to go beyond the linear phases, replacing them to higher order functions such as quadratic phases n → e(αn 2 ). This has given rise to the sub- ject of quadratic Fourier analysis, and more generally to higher order Fourier analysis. The classical results of Weyl on the equidistribution of poly- nomials (and their generalisations to other orbits on homogeneous spaces) can be interpreted through this perspective as foundational results in this subject. However, the modern theory of higher order Fourier analysis is very recent indeed (and still incomplete to some extent), beginning with the breakthrough work of Gowers [Go1998], [Go2001] and also heavily influenced by parallel work in ergodic the- ory, in particular the seminal work of Host and Kra [HoKr2005]. This area was also quickly seen to have much in common with ar- eas of theoretical computer science related to polynomiality testing, and in joint work with Ben Green and Tamar Ziegler [GrTa2010], [GrTa2008c], [GrTaZi2010b], applications of this theory were given to asymptotics for various linear patterns in the prime numbers. ix x Preface There are already several surveys or texts in the literature (e.g. [Gr2007], [Kr2006], [Kr2007], [Ho2006], [Ta2007], [TaVu2006b]) that seek to cover some aspects of these developments. In this text (based on a topics graduate course I taught in the spring of 2010), I attempt to give a broad tour of this nascent field. This text is not intended to directly substitute for the core papers in the subject (many of which are quite technical and lengthy), but focuses instead on basic foundational and preparatory material, and on the simplest illustrative examples of key results, and should thus hopefully serve as a companion to the existing literature on the subject. In accor- dance with this complementary intention of this text, we also present certain approaches to the material that is not explicitly present in the literature, such as the abstract approach to Gowers-type norms (Section 2.2) or the ultrafilter approach to equidistribution (Section 1.1.3). This text presumes a graduate-level familiarity with basic real analysis and measure theory, such as is covered in [Ta2011], [Ta2010], particularly with regard to the “soft” or “qualitative” side of the sub- ject. The core of the text is Chapter 1, which comprise the main lecture material. The material in Chapter 2 is optional to these lectures, ex- cept for the ultrafilter material in Section 2.1 which would be needed to some extent in order to facilitate the ultralimit analysis in Chapter 1. However, it is possible to omit the portions of the text involving ultrafilters and still be able to cover most of the material (though from a narrower set of perspectives). Acknowledgments I am greatly indebted to my students of the course on which this text was based, as well as many further commenters on my blog, including Sungjin Kim, William Meyerson, Joel Moreira, and Mads Sørensen. These comments, as well as the original lecture notes for this course, can be viewed online at terrytao.wordpress.com/category/teaching/254a-random-matrices The author is supported by a grant from the MacArthur Founda- tion, by NSF grant DMS-0649473, and by the NSF Waterman award. Chapter 1 Higher order Fourier analysis 1 2 1. Higher order Fourier analysis 1.1. Equidistribution of polynomial sequences in tori (Linear) Fourier analysis can be viewed as a tool to study an arbitrary function f on (say) the integers Z, by looking at how such a function correlates with linear phases such as n → e(ξn), where e(x) := e 2πix is the fundamental character, and ξ ∈ R is a frequency. These cor- relations control a number of expressions relating to f, such as the expected behaviour of f on arithmetic progressions n, n + r, n + 2r of length three. In this text we will be studying higher-order correlations, such as the correlation of f with quadratic phases such as n → e(ξn 2 ), as these will control the expected behaviour of f on more complex pat- terns, such as arithmetic progressions n, n+ r, n + 2r, n + 3r of length four. In order to do this, we must first understand the behaviour of exponential sums such as N  n=1 e(αn 2 ). Such sums are closely related to the distribution of expressions such as αn 2 mod 1 in the unit circle T := R/Z, as n varies from 1 to N. More generally, one is interested in the distribution of polynomials P : Z d → T of one or more variables taking values in a torus T; for instance, one might be interested in the distribution of the quadruplet (αn 2 , α(n + r) 2 , α(n + 2r) 2 , α(n + 3r) 2 ) as n, r both vary from 1 to N . Roughly speaking, once we understand these types of distributions, then the general machinery of quadratic Fourier analysis will then allow us to understand the distribution of the quadruplet (f(n), f(n+ r), f (n+2r), f(n+3r)) for more general classes of functions f ; this can lead for instance to an understanding of the distribution of arithmetic progressions of length 4 in the primes, if f is somehow related to the primes. More generally, to find arithmetic progressions such as n, n+r, n+ 2r, n + 3r in a set A, it would suffice to understand the equidistribu- tion of the quadruplet 1 (1 A (n), 1 A (n + r), 1 A (n + 2r), 1 A (n + 3r)) in 1 Here 1 A is the indicator function of A, defined by setting 1 A (n) equal to 1 when n ∈ A and equal to zero otherwise. [...]... periodic of period m, and definitely not equidistributed In the one-dimensional case d = 1, these are the only two possibilities But in higher dimensions, one can have a mixture of the two extremes, that exhibits irrational behaviour in some directions 8 1 Higher order Fourier analysis and periodic behaviour in others Consider for instance the two√ dimensional sequence n → ( 2n, 1 n) mod Z2 The first coordinate... |En∈[N ] f (xn )| > δ We introduce a summation parameter R ∈ N, and consider the Fej´r e partial Fourier series ˆ mR (k)f (k)e(k · x) FR f (x) := k∈Zd ˆ where f (k) are the Fourier coefficients ˆ f (k) := f (x)e(−k · x) dx Td and mR is the Fourier multiplier d 1− mR (k1 , , kd ) := j=1 |kj | R + Standard Fourier analysis shows that we have the convolution representation FR f (x) = f (y)KR (x − y) Td where... some absolute constant C, and if we need C to depend on additional parameters then we will indicate this by subscripts, e.g X d Y means that |X| ≤ Cd Y for some Cd depending only on d In 4 1 Higher order Fourier analysis the ultralimit theory we will use an analogue of asymptotic notation, which we will review later in this section 1.1.1 Asymptotic equidistribution theory Before we look at the single-scale... derivative sequence ∂h x : n → x(n + h) − x(n) is asymptotically equidistributed on N for all positive integers h Then xn is asymptotically equidistributed on N Similarly with N replaced by Z 10 1 Higher order Fourier analysis Proof We just prove the claim for N, as the claim for Z is analogous (and can in any case be deduced from the N case.) By Proposition 1.1.2, we need to show that for each non-zero k... P (nj ) → P (n0 ) We discussed recurrence for one-dimensional sequences x : n → x(n) It is also of interest to establish an analogous theory for multidimensional sequences, as follows 12 1 Higher order Fourier analysis Definition 1.1.10 A multidimensional sequence x : Zm → X is asymptotically equidistributed relative to a probability measure µ if, for every continuous, compactly supported function... polynomial map from Z2 to T k that maps (a, r) to (P (a), , P (a + (k − 1)r)) One can also use the one-dimensional theory by freezing a and only looking at the equidistribution in r.) 14 1 Higher order Fourier analysis 1.1.2 Single-scale equidistribution theory We now turn from the asymptotic equidistribution theory to the equidistribution theory at a single scale N Thus, instead of analysing the... single-scale equidistribution We begin with the Weyl criterion Proposition 1.1.13 (Single-scale Weyl equidistribution criterion) Let x1 , x2 , , xN be a sequence in Td , and let 0 < δ < 1 16 1 Higher order Fourier analysis (i) If x1 , , xN is δ-equidistributed, and k ∈ Zd \{0} has magnitude |k| ≤ δ −c , then one has |En∈[N ] e(k · xn )| δc d if c > 0 is a small enough absolute constant (ii) Conversely,... Let ϕ : R → R be a compactly supported, piecewise continuous function with only finitely many pieces Show that for any f ∈ C(X) one has ∞ 1 lim ϕ(n/N )f (x(n)) = f dµ ϕ(t) dt N →∞ N X 0 n∈N 6 1 Higher order Fourier analysis and for any open U whose boundary has measure zero, one has 1 N →∞ N ∞ lim ϕ(n/N ) = µ(U ) n∈N:x(n)∈U ϕ(t) dt 0 In this set of notes, X will be a torus (i.e a compact connected abelian... δ-equidistributed, then there exists k ∈ Zd \{0} with magnitude |k| d δ −Cd , and a rational a of height Od (δ −Cd ), such that |En∈[N ] e(k · xn )e(an)| for some Cd depending on d d δ Cd 18 1 Higher order Fourier analysis This gives a version of Exercise 1.1.5: Exercise 1.1.19 Let α, β ∈ Td , let N ≥ 1, and let 0 < δ < 1 Suppose that the linear sequence (αn + β)N is not totally δn=1 equidistributed Show... allow all implied constants to depend on s From Exercise 1.1.21, we already can find a positive k with k = O(δ −O(1) ) such that kαs T δ −O(1) /N s We now partition [N ] into arithmetic 20 1 Higher order Fourier analysis progressions of spacing k and length N ∼ δ C N for some sufficiently large C; then by the pigeonhole principle, we see that P fails to be totally δ O(1) -equidistributed on one of these . Waterman award. Chapter 1 Higher order Fourier analysis 1 2 1. Higher order Fourier analysis 1.1. Equidistribution of polynomial sequences in tori (Linear) Fourier analysis can be viewed as a. them to higher order functions such as quadratic phases n → e(αn 2 ). This has given rise to the sub- ject of quadratic Fourier analysis, and more generally to higher order Fourier analysis. The. Ultralimit analysis and quantitative algebraic geometry156 §2.2. Higher order Hilbert spaces 180 §2.3. The uncertainty principle 195 Bibliography 215 Index 221 vii Preface Traditionally, Fourier analysis

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