# Terence Tao's research while affiliated with University of Turku and other places

## Publications (305)

Article
Full-text available
Define the Collatz map ${\operatorname {Col}} \colon \mathbb {N}+1 \to \mathbb {N}+1$ on the positive integers $\mathbb {N}+1 = \{1,2,3,\dots \}$ by setting ${\operatorname {Col}}(N)$ equal to $3N+1$ when N is odd and $N/2$ when N is even, and let ${\operatorname {Col}}_{\min }(N) := \inf _{n \in \mathbb {N}} {\operatorname {Col}}^n(N)$ denote the...
Preprint
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We study higher uniformity properties of the M\"obius function $\mu$, the von Mangoldt function $\Lambda$, and the divisor functions $d_k$ on short intervals $(X,X+H]$ with $X^{\theta+\varepsilon} \leq H \leq X^{1-\varepsilon}$ for a fixed constant $0 \leq \theta < 1$ and any $\varepsilon>0$. More precisely, letting $\Lambda^\sharp$ and $d_k^\sharp... Article Singmaster’s conjecture asserts that every natural number greater than one occurs at most a bounded number of times in Pascal’s triangle; that is, for any natural number$t \geq 2$, the number of solutions to the equation$\binom{n}{m} = t$for natural numbers$1 \leq m \lt n$is bounded. In this paper we establish this result in the interior regio... Preprint Assuming that Siegel zeros exist, we prove a hybrid version of the Chowla and Hardy--Littlewood prime tuples conjectures. Thus, for an infinite sequence of natural numbers$x$, and any distinct integers$h_1,\dots,h_k,h'_1,\dots,h'_\ell$, we establish an asymptotic formula for $$\sum_{n\leq x}\Lambda(n+h_1)\cdots \Lambda(n+h_k)\lambda(n+h_{1}')\cdo... Article Full-text available A bipartite graph H = \left (V_1, V_2; E \right ) with \lvert V_1\rvert + \lvert V_2\rvert = n is semilinear if V_i \subseteq \mathbb {R}^{d_i} for some d_i and the edge relation E consists of the pairs of points (x_1, x_2) \in V_1 \times V_2 satisfying a fixed Boolean combination of s linear equalities and inequalities in d_1 + d_2 var... Preprint We establish quantitative bounds on the U^k[N] Gowers norms of the M\"obius function \mu and the von Mangoldt function \Lambda for all k, with error terms of shape O((\log\log N)^{-c}). As a consequence, we obtain quantitative bounds for the number of solutions to any linear system of equations of finite complexity in the primes, with the... Preprint Full-text available Singmaster's conjecture asserts that every natural number greater than one occurs at most a bounded number of times in Pascal's triangle; that is, for any natural number t \geq 2, the number of solutions to the equation \binom{n}{m} = t for natural numbers 1 \leq m < n is bounded. In this paper we establish this result in the interior region... Article Singmaster's conjecture asserts that every natural number greater than one occurs at most a bounded number of times in Pascal's triangle; that is, for any natural number t ≥ 2, the number of solutions to the equation (n / m) = t for natural numbers 1 ≤ m < n is bounded. In this paper we establish this result in the interior region exp(log^(2/3+ε)n)... Preprint Full-text available A bipartite graph H = \left(V_1, V_2; E \right) with |V_1| + |V_2| = n is semilinear if V_i \subseteq \mathbb{R}^{d_i} for some d_i and the edge relation E consists of the pairs of points (x_1, x_2) \in V_1 \times V_2 satisfying a fixed Boolean combination of s linear equalities and inequalities in d_1 + d_2 variables for some s.... Preprint Let \lambda denote the Liouville function. We show that, as X \rightarrow \infty,$$\int_{X}^{2X} \sup_{\substack{P(Y)\in \mathbb{R}[Y]\\ deg(P)\leq k}} \Big | \sum_{x \leq n \leq x + H} \lambda(n) e(-P(n)) \Big |\ dx = o ( X H)$$for all fixed k and X^{\theta} \leq H \leq X with 0 < \theta < 1 fixed but arbitrarily small. Previously this... Article 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)... Preprint We study homogenization of iterated randomized singular integrals and homeomorphic solutions to the Beltrami differential equation with a random Beltrami coefficient. More precisely, let (F_j)_{j \geq 1} be a sequence of normalized homeomorphic solutions to the planar Beltrami equation \overline{\partial} F_j (z)=\mu_j(z,\omega) \partial F_j(z),... Article Full-text available Let λ denote the Liouville function. We show that as X→∞, ∫^(2X)X supα∣∑x< n ≤ x+H λ(n)e(−αn)∣dx = o(XH) for all H ≥ X^θ with θ > 0 fixed but arbitrarily small. Previously, this was only known for θ > 5/8. For smaller values of θ this is the first “non-trivial” case of local Fourier uniformity on average at this scale. We also obtain the analogous... Article Full-text available The original version of this article unfortunately contains a mistake. Article Full-text available We study the existence of various sign and value patterns in sequences defined by multiplicative functions or related objects. For any set A whose indicator function is ‘approximately multiplicative’ and uniformly distributed on short intervals in a suitable sense, we show that the density of the pattern n+1\in A , n+2\in A , n+3\in A is po... Preprint We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we obtain heuristic upper and lower bounds for the size of the largest prime gap in the interval [1,x]. Our res... Article Full-text available We study the problem of obtaining asymptotic formulas for the sums \sum_{X < n \leq 2X} d_k(n) d_l(n+h) and \sum_{X < n \leq 2X} \Lambda(n) d_k(n+h), where \Lambda is the von Mangoldt function, d_k is the k^{\operatorname{th}} divisor function, X is large and k \geq l \geq 2 are real numbers. We show that for almost all h \in [-H, H]... Article Despite much effort, the question of whether the Navier–Stokes equations allow solutions that develop singularities in finite time remains unresolved. Terence Tao discusses the problem, and possible routes to a solution. Preprint We study the existence of various sign and value patterns in sequences defined by multiplicative functions or related objects. For any set A whose indicator function is 'approximately multiplicative' and uniformly distributed on short intervals in a suitable sense, we show that the asymptotic density of the pattern n+1\in A, n+2\in A, n+3\in... Preprint Let \lambda denote the Liouville function. We show that as X \rightarrow \infty,$$ \int_{X}^{2X} \sup_{\alpha} \left | \sum_{x < n \leq x + H} \lambda(n) e(-\alpha n) \right | dx = o ( X H) $$for all H \geq X^{\theta} with \theta > 0 fixed but arbitrarily small. Previously, this was only known for \theta > 5/8. For smaller values of \t... Article Let G = (G, +) be a compact connected abelian group, and let μG denote its probability Haar measure. A theorem of Kneser (generalising previous results of Macbeath, Raikov, and Shields) establishes the bound μG(A + B) ≥ min(μG(A) + μG(B), 1) whenever A and B are compact subsets of G, and A + B:= {a + b: a ∈ A, b ∈ B} denotes the sumset of A and B.... Preprint We study the asymptotic behaviour of higher order correlations$$ \mathbb{E}_{n \leq X/d} g_1(n+ah_1) \cdots g_k(n+ah_k)$$as a function of the parameters a and d, where g_1,\dots,g_k are bounded multiplicative functions, h_1,\dots,h_k are integer shifts, and X is large. Our main structural result asserts, roughly speaking, that such corr... Chapter Let pn denote the n-th prime, and for any $$k \geqslant 1$$ and sufficiently large X, define the quantity$$\displaystyle G_k(X) := \max _{p_{n+k} \leqslant X} \min ( p_{n+1}-p_n, \dots , p_{n+k}-p_{n+k-1} ), $$which measures the occurrence of chains of k consecutive large gaps of primes. Recently, with Green and Konyagin, the authors showed that... Article To each prime p, let I_p \subset \mathbb{Z}/p\mathbb{Z} denote a collection of at most C_0 residue classes modulo p, whose cardinality |I_p| is equal to 1 on the average. We show that for sufficiently large x, the sifted set \{ n \in \mathbb{Z}: n \pmod{p} \not \in I_p \hbox{ for all }p \leq x\} contains gaps of size x (\log x)^{1/\... Article Define r_{4}(N) to be the largest cardinality of a set A\subset \{1,\ldots ,N\} that does not contain four elements in arithmetic progression. In 1998, Gowers proved that$$\begin{eqnarray}r_{4}(N)\ll N(\log \log N)^{-c}\end{eqnarray}$$for some absolute constant c>0 . In 2005, the authors improved this to$$\begin{eqnarray}r_{4}(N)\ll N\text... Article A famous conjecture of Chowla states that the Liouville function$\lambda(n)$has negligible correlations with its shifts. Recently, the authors established a weak form of the logarithmically averaged Elliott conjecture on correlations of multiplicative functions, which in turn implied all the odd order cases of the logarithmically averaged Chowla... Article Let$g_0,\dots,g_k: {\bf N} \to {\bf D}$be$1$-bounded multiplicative functions, and let$h_0,\dots,h_k \in {\bf Z}$be shifts. We consider correlation sequences$f: {\bf N} \to {\bf Z}$of the form $$f(a):= \widetilde{\lim}_{m \to \infty} \frac{1}{\log \omega_m} \sum_{x_m/\omega_m \leq n \leq x_m} \frac{g_0(n+ah_0) \dots g_k(n+ah_k)}{n}$$ where... Article Given a smooth potential function$V : \mathbf{R}^m \to \mathbf{R}$, one can consider the ODE$\partial_t^2 u = -(\nabla V)(u)$describing the trajectory of a particle$t \mapsto u(t)$in the potential well$V$. We consider the question of whether the dynamics of this family of ODE are \emph{universal} in the sense that they contain (as embedded co... Article We show that the expected asymptotic for the sums$\sum_{X < n \leq 2X} \Lambda(n) \Lambda(n+h)$,$\sum_{X < n \leq 2X} d_k(n) d_l(n+h)$, and$\sum_{X < n \leq 2X} \Lambda(n) d_k(n+h)$hold for almost all$h \in [-H,H]$, provided that$X^{8/33+\varepsilon} \leq H \leq X^{1-\varepsilon}$, with an error term saving on average an arbitrary power of th... Article Define$r_4(N)$to be the largest cardinality of a set$A \subset \{1,\dots,N\}$which does not contain four elements in arithmetic progression. In 1998 Gowers proved that $r_4(N) \ll N(\log \log N)^{-c}$ for some absolute constant$c>0$. In 2005, the authors improved this to $r_4(N) \ll N e^{-c\sqrt{\log\log N}}.$ In this paper we further im... Article Full-text available Gaps (or spacings) between consecutive eigenvalues are a central topic in random matrix theory. The goal of this paper is to study the tail distribution of these gaps in various random matrix models. We give the first repulsion bound for random matrices with discrete entries and the first super-polynomial bound on the probability that a random grap... Article Full-text available The "square peg problem" or "inscribed square problem" of Toeplitz asks if every simple closed curve in the plane inscribes a (non-degenerate) square, in the sense that all four vertices of that square lie on the curve. By a variety of arguments of a "homological" nature, it is known that the answer to this question is positive if the curve is suff... Chapter In the previous chapter, we defined the real numbers as formal limits of rational (Cauchy) sequences, and we then defined various operations on the real numbers. However, unlike our work in constructing the integers (where we eventually replaced formal differences with actual differences) and rationals (where we eventually replaced formal quotients... Chapter Modern analysis, like most of modern mathematics, is concerned with numbers, sets, and geometry. We have already introduced one type of number system, the natural numbers. Chapter In this text, we will review the material you have learnt in high school and in elementary calculus classes, but as rigorously as possible. To do so we will have to begin at the very basics - indeed, we will go back to the concept of numbers and what their properties are. Of course, you have dealt with numbers for over ten years and you know how to... Chapter We can now begin the rigorous treatment of calculus in earnest, starting with the notion of a derivative. We can now define derivatives analytically, using limits, in contrast to the geometric definition of derivatives, which uses tangents. The advantage of working analytically is that (a) we do not need to know the axioms of geometry, and (b) thes... Chapter Now that we have developed a reasonable theory of limits of sequences, we will use that theory to develop a theory of infinite series. Chapter In Chapter 2 we built up most of the basic properties of the natural number system, but we have reached the limits of what one can do with just addition and multiplication. We would now like to introduce a new operation, that of subtraction, but to do that properly we will have to pass from the natural number system to a larger number system, that... Chapter We now return to the study of set theory, and specifically to the study of cardinality of sets which are infinite (i.e., sets which do not have cardinality n for any natural number n), a topic which was initiated in Section 3.6. Chapter To review our progress to date, we have rigorously constructed three fundamental number systems: the natural number system N, the integers Z, and the rationals Q 1. We defined the natural numbers using the five Peano axioms, and postulated that such a number system existed; this is plausible, since the natural numbers correspond to the very intuiti... Chapter In previous chapters we have been focusing primarily on sequences. A sequence $$\left( {a_{n} } \right)_{n = 0}^{\infty }$$ can be viewed as a function from N to R, i.e., an object which assigns a real number a n to each natural number n. We then did various things with these functions from N to R, such as take their limit at infinity (if the fun... Chapter In the previous chapter we reviewed differentiation—one of the two pillars of single variable calculus. The other pillar is, of course, integration, which is the focus of the current chapter. More precisely, we will turn to the definite integral, the integral of a function on a fixed interval, as opposed to the indefinite integral, otherwise known... Chapter In Definition 6.1.5 we defined what it meant for a sequence $$\left( {x_{n} } \right)_{n = m}^{\infty }$$ of real numbers to converge to another real number x; indeed, this meant that for every ε > 0, there exists an N ≥ m such that|x − x n | ≤ ε for all n ≥ N. When this is the case, we write limn→∞ x n = x. Chapter In the previous two chapters we have seen what it means for a sequence $$\left( {x^{(n)} } \right)_{n = 1}^{\infty }$$ of points in a metric space $$\left( {X,d_{X} } \right)$$ to converge to a limit x; it means that $$\lim_{n \to \infty } d_{X} \left( {x^{(n)} ,x} \right) < \varepsilon$$ or equivalently that for every $$\varepsilon > 0$$ t... Chapter We now discuss an important subclass of series of functions, that of power series. As in earlier chapters, we begin by introducing the notion of a formal power series, and then focus in later sections on when the series converges to a meaningful function, and what one can say about the function obtained in this manner. Chapter In the previous two chapters, we discussed the issue of how certain functions (for instance, compactly supported continuous functions) could be approximated by polynomials. Later, we showed how a different class of functions (real analytic functions) could be written exactly (not approximately) as an infinite polynomial, or more precisely a power s... Chapter In the previous chapter we discussed differentiation in several variable calculus. It is now only natural to consider the question of integration in several variable calculus. Chapter In the previous chapter we studied a single metric space (X, d), and the various types of sets one could find in that space. While this is already quite a rich subject, the theory of metric spaces becomes even richer, and of more importance to analysis, when one considers not just a single metric space, but rather pairs (X, d X ) and (Y, d Y ) of m... Chapter We shall now switch to a different topic, namely that of differentiation in several variable calculus. More precisely, we shall be dealing with maps f : R n → R m from one Euclidean space to another, and trying to understand what the derivative of such a map is. Chapter In Chapter 11, we approached the Riemann integral by first integrating a particularly simple class of functions, namely the piecewise constant functions. Among other things, piecewise constant functions only attain a finite number of values (as opposed to most functions in real life, which can take an infinite number of values). Once one learns how... Article Full-text available Let${\it\lambda}$and${\it\mu}$denote the Liouville and Möbius functions, respectively. Hildebrand showed that all eight possible sign patterns for$({\it\lambda}(n),{\it\lambda}(n+1),{\it\lambda}(n+2))$occur infinitely often. By using the recent result of the first two authors on mean values of multiplicative functions in short intervals, we s... Article Full-text available Let$P_1,\dots,P_k \colon {\bf Z} \to {\bf Z}$be polynomials of degree at most$d$for some$d \geq 1$, with the degree$d$coefficients all distinct, and admissible in the sense that for every prime$p$, there exists integers$n,m$such that$n+P_1(m),\dots,n+P_k(m)$are all not divisible by$p$. We show that there exist infinitely many natural n... Article Full-text available Concatenation theorems for anti-Gowers-uniform functions and Host-Kra characteristic factors, Discrete Analysis 2016:13, 61 pp. It is a straightforward exercise to show that if$f$is a function of two integer variables, and$f(x,y)$is a polynomial of degree$m$in$x$when$y$is fixed and degree$n$in$y$when$x$is fixed, then$f$is a polyn... Article Full-text available We discuss several questions concerning sum-free sets in groups, raised by Erd\H{o}s in his survey "Extremal problems in number theory" (Proceedings of the Symp. Pure Math. VIII AMS) published in 1965. Among other things, we give a characterization for large sets$A$in an abelian group$G$which do not contain a subset$B$of fixed size$k$such t... Article Full-text available Sum-avoiding sets in groups, Discrete Analysis 2016:15, 27 pp. Let$A$be a subset of an Abelian group$G$. A subset$B\subset A$is called _sum-avoiding in$A$_ if no two elements of$B$add up to an element of$A$. Write$\phi(A)$for the size of the largest sum-avoiding subset of$A$. If$G=\mathbb Z$and$|A|=n$, then it is known that$\phi(A)...
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We consider the global regularity problem for nonlinear wave systems $$\Box u = f(u)$$ on Minkowski spacetime ${\bf R}^{1+d}$ with d'Alambertian $\Box := -\partial_t^2 + \sum_{i=1}^d \partial_{x_i}^2$, where the field $u \colon {\bf R}^{1+d} \to {\bf R}^m$ is vector-valued, and the nonlinearity $f \colon {\bf R}^m \to {\bf R}^m$ is a smooth funct...
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Let $p_n$ denote the $n$-th prime, and for any $k \geq 1$ and sufficiently large $X$, define the quantity $$G_k(X) := \max_{p_{n+k} \leq X} \min( p_{n+1}-p_n, \dots, p_{n+k}-p_{n+k-1} ),$$ which measures the occurrence of chains of $k$ consecutive large gaps of primes. Recently, with Green and Konyagin, the authors showed that $G_1(X) \gg \frac{... Article Let \lambda and \mu denote the Liouville and M\"obius functions respectively. Hildebrand showed that all eight possible sign patterns for (\lambda(n), \lambda(n+1), \lambda(n+2)) occur infinitely often. By using the recent result of the first two authors on mean values of multiplicative functions in short intervals, we strengthen Hildebrand's... Article Using an ergodic inverse theorem obtained in our previous paper, we obtain limit formulae for multiple ergodic averages associated with the action of (Formula Presented.). From this we deduce multiple Khintchine-type recurrence results analogous to those for ℤ-systems obtained by Bergelson, Host, and Kra, and also present some new counterexamples i... Article For any natural number k, consider the k-linear Hilbert transform  H_k( f_1,\dots,f_k )(x) := \operatorname{p.v.} \int_{\bf R} f_1(x+t) \dots f_k(x+kt)\ \frac{dt}{t} for test functions f_1,\dots,f_k: {\bf R} \to {\bf C}. It is conjectured that H_k maps L^{p_1}({\bf R}) \times \dots \times L^{p_k}({\bf R}) \to L^p({\bf R}) whenever 1... Article Let \lambda denote the Liouville function. A well known conjecture of Chowla asserts that for any distinct natural numbers h_1,\dots,h_k, one has \sum_{1 \leq n \leq X} \lambda(n+h_1) \dotsm \lambda(n+h_k) = o(X) as X \to \infty. This conjecture remains unproven for any h_1,\dots,h_k with k \geq 2. In this paper, using the recent result... Article Full-text available Let p n p_n denote the n n th prime. We prove that \[ max p n ⩽ X ( p n + 1 − p n ) ≫ log ⁡ X log ⁡ log ⁡ X log ⁡ log ⁡ log ⁡ log ⁡ X log ⁡ log ⁡ log ⁡ X \max _{p_{n} \leqslant X} (p_{n+1}-p_n) \gg \frac {\log X \log \log X\log \log \log \log X}{\log \log \log X}$ for sufficiently large X X , improving upon recent bounds of the first, second, thi...
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Let $M_n = (\xi_{ij})_{1 \leq i,j \leq n}$ be a real symmetric random matrix in which the upper-triangular entries $\xi_{ij}, i<j$ and diagonal entries $\xi_{ii}$ are independent. We show that with probability tending to 1, $M_n$ has no repeated eigenvalues. As a corollary, we deduce that the Erd{\H o}s-Renyi random graph has simple spectrum asympt...
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For each prime $p$, let $n(p)$ denote the least quadratic nonresidue modulo $p$. Vinogradov conjectured that $n(p) = O(p^\eps)$ for every fixed $\eps>0$. This conjecture follows from the generalised Riemann hypothesis, but remains open in general. In this paper we show that Vinogradov's conjecture follows from the Elliott-Halberstam conjecture on t...
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In a previous paper of the authors, we showed that for any polynomials $P_1,\dots,P_k \in \Z[\mathbf{m}]$ with $P_1(0)=\dots=P_k(0)$ and any subset $A$ of the primes in $[N] = \{1,\dots,N\}$ of relative density at least $\delta>0$, one can find a "polynomial progression" $a+P_1(r),\dots,a+P_k(r)$ in $A$ with $0 < |r| \leq N^{o(1)}$, if $N$ is suffi...
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Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdos, we show that $$G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2},$$ where $f(X)$ is a function tending to infinity with $X$. Our proof combines existing arguments with a random construction covering a...
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The Navier-Stokes equation on the Euclidean space $\mathbf{R}^3$ can be expressed in the form $\partial_t u = \Delta u + B(u,u)$, where $B$ is a certain bilinear operator on divergence-free vector fields $u$ obeying the cancellation property $\langle B(u,u), u\rangle=0$ (which is equivalent to the energy identity for the Navier-Stokes equation). In...
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This chapter introduces two sets of theorems based on a study of the Littlewood–Paley boundedness properties of a tri-linear operator. It generalizes and proves similar estimates for multi-linear multipliers whose symbols are given by characteristic functions of simplexes of arbitrary length. The chapter introduces the so-called AKNS systems, which...
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This is an erratum to 'On the quantitative distribution of polynomial nilsequences' [GT]. The proof of Theorem 8.6 of that paper, which claims a distribution result for multiparameter polynomial sequences on nilmanifolds, was incorrect. We provide two fixes for this issue here. First, we deduce the "equal sides" case $N_1 = \dots = N_t = N$ of [GT,...
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We show that random Cayley graphs of finite simple (or semisimple) groups of Lie type of fixed rank are expanders. The proofs are based on the Bourgain-Gamburd method and on the main result of our companion paper, establishing strongly dense subgroups in simple algebraic groups.
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In this paper, we establish some local universality results concerning the correlation functions of the zeroes of random polynomials with independent coefficients. More precisely, consider two random polynomials $f =\sum _{i=1}^n c_i \xi _i z^i$ and $\tilde {f} =\sum _{i=1}^n c_i \tilde {\xi }_i z^i$, where the ξi and $\tilde {\xi }_i$ are iid rand...
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We establish a version of the Furstenberg-Katznelson multi-dimensional Szemer\'edi in the primes ${\mathcal P} := \{2,3,5,\ldots\}$, which roughly speaking asserts that any dense subset of ${\mathcal P}^d$ contains constellations of any given shape. Our arguments are based on a weighted version of the Furstenberg correspondence principle, relative...
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Using an ergodic inverse theorem obtained in our previous paper, we obtain limit formulae for multiple ergodic averages associated with the action of $\F_{p}^{\omega}$. From this we deduce multiple Khintchine-type recurrence results analogous to those for $\Z$-systems obtained by Bergelson, Host, and Kra, and also present some new counterexamples i...
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Let A be a subset of a group G = (G,.). We will survey the theory of sets A with the property that |A.A| <= K|A|, where A.A = {a_1 a_2 : a_1, a_2 in A}. The case G = (Z,+) is the famous Freiman--Ruzsa theorem.
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We establish a new mixing theorem for quasirandom groups (finite groups with no low-dimensional unitary representations) G which, informally speaking, asserts that if g, x are drawn uniformly at random from G, then the quadruple (g, x, gx, xg) behaves like a random tuple in G 4, subject to the obvious constraint that gx and xg are conjugate to each...
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Let $P: \F \times \F \to \F$ be a polynomial of bounded degree over a finite field $\F$ of large characteristic. In this paper we establish the following dichotomy: either $P$ is a moderate asymmetric expander in the sense that $|P(A,B)| \gg |\F|$ whenever $A, B \subset \F$ are such that $|A| |B| \geq C |\F|^{2-1/8}$ for a sufficiently large $C$, o...
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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|| Us+1[N] ≥ δ, then there is a bounded complexitys-step nilsequence F(g(n)Γ) that correlates with f, where the bounds on the complexity and correlation depend only on s and δ...
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Let P be a set of n points in the plane, not all on a line. We show that if n is large then there are at least n/2 ordinary lines, that is to say lines passing through exactly two points of P. This confirms, for large n, a conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for this problem, as well as all sets having fewer t...
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It is a classical result of Ginibre that the normalized bulk $k$-point correlation functions of a complex $n \times n$ gaussian matrix with independent entries of mean zero and unit variance are asymptotically given by the determinantal point process on $\C$ with kernel $K_\infty(z,w) := \frac{1}{\pi} e^{-|z|^2/2 - |w|^2/2 + z \bar{w}}$ in the limi...
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Let p > 4 be a prime. We show that the largest subset of F_p^n with no 4-term arithmetic progressions has cardinality << N(log N)^{-c}, where c = 2^{-22} and N := p^n. A result of this type was claimed in a previous paper by the authors and published in Proc. London Math. Society. Unfortunately the proof had a gap, and we issue an erratum for that...
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We show that the distribution of (a suitable rescaling of) a single eigenvalue gap $\lambda_{i+1}(M_n)-\lambda_i(M_n)$ of a random Wigner matrix ensemble in the bulk is asymptotically given by the Gaudin-Mehta distribution, if the Wigner ensemble obeys a finite moment condition and matches moments with the GUE ensemble to fourth order. This is new...
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We prove that every odd number $N$ greater than 1 can be expressed as the sum of at most five primes, improving the result of Ramar\'e that every even natural number can be expressed as the sum of at most six primes. We follow the circle method of Hardy-Littlewood and Vinogradov, together with Vaughan's identity; our additional techniques, which ma...
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In this paper, we survey some recent progress on rigorously establishing the universality of various spectral statistics of Wigner Hermitian random matrix ensembles, focusing on the Four Moment Theorem and its refinements and applications, including the universality of the sine kernel and the Central limit theorem of several spectral parameters. We...
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The nonlinear Fourier transform discussed in these notes is the map from the potential of a one dimensional discrete Dirac operator to the transmission and reflection coefficients thereof. Emphasis is on this being a nonlinear variant of the classical Fourier series, and on nonlinear analogues of classical analytic facts about Fourier series. These...
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Let $W_n= \frac{1}{\sqrt n} M_n$ be a Wigner matrix whose entries have vanishing third moment, normalized so that the spectrum is concentrated in the interval $[-2,2]$. We prove a concentration bound for $N_I = N_I(W_n)$, the number of eigenvalues of $W_n$ in an interval $I$. Our result shows that $N_I$ decays exponentially with standard deviation...
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We prove that a K-approximate subgroup of an arbitrary torsion-free nilpotent group can be covered by a bounded number of cosets of a nilpotent subgroup of bounded rank, where the bounds are explicit and depend only on K. The result can be seen as a nilpotent analogue to Freiman's dimension lemma.
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We survey some recent progress on rigorously establishing the universality of various spectral statistics of Wigner random matrix ensembles, focusing in particular on the Four Moment Theorem and its applications.
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We establish a central limit theorem for the log-determinant $\log|\det(M_n)|$ of a Wigner matrix $M_n$, under the assumption of four matching moments with either the GUE or GOE ensemble. More specifically, we show that this log-determinant is asymptotically distributed like $N(\log \sqrt{n!} - 1/2 \log n, 1/2 \log n)_\R$ when one matches moments w...
Article
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....
Article
In this paper we establish a number of implications between various qualitative and quantitative versions of the global regularity problem for the Navier-Stokes equations, in the periodic, smooth finite energy, smooth $H^1$, Schwartz, or mild $H^1$ categories, and with or without a forcing term. In particular, we show that if one has global well-po...
Article
This is an announcement of the proof of the inverse conjecture for the Gowers Us+1[N]-norm for all s ≥ 3; this is new for s ≥ 4, the cases s = 1, 2, 3 having been previously established. More precisely we outline a proof that if f: [N] → [-1, 1] is a function with kfkUs+1[N] ≥ δ then there is a bounded-complexity s-step nilsequence F(g(n) Γ) which...
Article
A corollary of Kneser's theorem, one sees that any finite non-empty subset $A$ of an abelian group $G = (G,+)$ with $|A + A| \leq (2-\eps) |A|$ can be covered by at most $\frac{2}{\eps}-1$ translates of a finite group $H$ of cardinality at most $(2-\eps)|A|$. Using some arguments of Hamidoune, we establish an analogue in the noncommutative setting....
Chapter
We give a quick tour through some topics in analytic prime number theory, focusing in particular on the strange mixture of order and chaos in the primes. For instance, while primes do obey some obvious patterns (e.g. they are almost all odd), and have a very regular asymptotic distribution (the prime number theorem), we still do not know a determin...

## Citations

... It is natural to consider the following hybrid conjecture, which, following [10] and [20], we call the Hardy-Littlewood-Chowla conjecture. Conjecture 1.1 (Hardy-Littlewood-Chowla). Let , ℓ ≥ 0, and let ℎ 1 , . . . ...
... A number of authors presented models that describe random walks to simulate the function [5][6][7]. Tao recently proved that "Almost all orbits of the Collatz map attain almost bounded values" [8]. The Collatz system may be described as a physical system as it behaves similarly to a physical system operating under a feedback design system using a sliding mode control process [9]. ...
... The Zarankiewicz problem [40] asks for the maximum number of edges in a bipartite graph on n + n vertices containing no copy of K s,t , i.e. the complete bipartite graph with vertex classes of size s and t. A geometric variant of this problem was recently introduced by Basit, Chernikov, Starchenko, Tao, and Tran [7]. Given a set P and family of sets F, the incidence graph of (P, F) is the bipartite graph with vertex classes P and F, where p ∈ P and F ∈ F are joined by an edge if p ∈ F . ...
... The rest of the paper is therefore devoted to the proofs of Theorem 1.6 and Theorem 1.2(ii). Theorem 1.6 improves on the range H ≥ exp((log X) 5/8+ε ) which follows (under a slightly different pretentiousness hypothesis) from Fourier uniformity bounds of Matomäki, Radziwi l l, Tao, Ziegler and the second author [18,Theorem 1.8]. See [11,Theorem 1.8] for the details of this implication 2 . ...
... The cost of this improvement was in the dependency on ε of the distortion constant of the bi-Lipschitz embedding. Tao [264] considered the embedding properties of the Heisenberg group in detail, proving that the ε-snowflakings of H can be embedded into R n with n independent of ε and with the optimal distortion constant. ...
... Despite recent progress due to Tao and Teräväinen [36,37,38], concerning the case of polynomials that factor into linear factors, this conjecture remains wide open for any f that is irreducible and of degree at least 2. In the spirit of Theorem 1.2, we can prove the polynomial Chowla conjecture for almost all polynomials, improving on a qualitative result due to Teräväinen [39,Thm. 2.12]. ...
... See [18,31,40] and references therein. Notable progress has also been made by Matomäki, Radziwiłł, Tao, Teräväinen, and many others, in case F splits as a product of linear factors, see [43,Introduction]. ...
... This result has been improved by Matomäki, Radziwill and Tao [8], who prove that these eight values appear with a positive lower density. Similar results and conjectures are stated for the Möbius function, or for the number of prime factors modulo 3 (see [10]). ...
... Tao's recent research [4] focus on the comparison of the magnitude of the convection term and the viscous term. When flow speed greatly exceeds the wave number, convection terms dominates; it might be possible that the flow velocity and wave number go to infinity. ...
... This Cauchy-Davenport theorem was followed by important contributions concerning sums of subsets of groups, including Z itself, by Mann [40,41], Kneser [35], Vosper [58,59], Erdős and Heilbronn [16], Freiman [17][18][19], Plünnecke [47], Ruzsa [49], and others until the 1990s, when the subject really took off (see e.g. [2,4,31,39,46,50,51,54,56]). For various The Cauchy-Davenport theorem asserts that if A and B are non-empty subsets of Z p with |A| + |B| p, then |A + B| |A| + |B| − 1. ...