arXiv subjects

Terence Tao

Publications and source records attributed to Terence Tao.

At least 127 records · Page 7Linked to original sources

arXiv2016

Finite time blowup for a supercritical defocusing nonlinear Schrödinger system

We consider the global regularity problem for defocusing nonlinear Schrödinger systems $$ i \partial_t + Δu = (\nabla_{{\bf R}^m} F)(u) + G $$ on Galilean spacetime ${\bf R} \times {\bf R}^d$, where the field $u\colon {\bf R}^{1+d} \to {\bf C}^m$ is vector-valued, $F\colon {\bf C}^m \to {\bf R}$ is a smooth potential which is positive, phase-rotation-invariant, and homogeneous of order $p+1$ outside of the unit ball for some exponent $p >1$, and $G: {\bf R} \times {\bf R}^d \to {\bf C}^m$ is a smooth, compactly supported forcing term. This generalises the scalar defocusing nonlinear Schrödinger (NLS) equation, in which $m=1$ and $F(v) = \frac{1}{p+1} |v|^{p+1}$. In this paper we study the supercritical case where $d \geq 3$ and $p > 1 + \frac{4}{d-2}$. We show that in this case, there exists a smooth potential $F$ for some sufficiently large $m$, positive and homogeneous of order $p+1$ outside of the unit ball, and a smooth compactly choice of initial data $u(0)$ and forcing term $G$ for which the solution develops a finite time singularity. In fact the solution is locally discretely self-similar with respect to parabolic rescaling of spacetime. This demonstrates that one cannot hope to establish a global regularity result for the scalar defocusing NLS unless one uses some special property of that equation that is not shared by these defocusing nonlinear Schrödinger systems. As in a previous paper of the author considering the analogous problem for the nonlinear wave equation, the basic strategy is to first select the mass, momentum, and energy densities of $u$, then $u$ itself, and then finally design the potential $F$ in order to solve the required equation.

math.AP↗

arXiv2016

Finite time blowup for Lagrangian modifications of the three-dimensional Euler equation

In the language of differential geometry, the incompressible inviscid Euler equations can be written in vorticity-vector potential form as \begin{align*} \partial_t ω+ {\mathcal L}_u ω&= 0\\ u &= δ\tilde η^{-1} Δ^{-1} ω\end{align*} where $ω$ is the vorticity $2$-form, ${\mathcal L}_u$ denotes the Lie derivative with respect to the velocity field $u$, $Δ$ is the Hodge Laplacian, $δ$ is the codifferential (the negative of the divergence operator), and $\tilde η^{-1}$ is the canonical map from $2$-forms to $2$-vector fields induced by the Euclidean metric $η$. In this paper we consider a generalisation of these Euler equations in three spatial dimensions, in which the vector potential operator $\tilde η^{-1} Δ^{-1}$ is replaced by a more general operator $A$ of order $-2$; this retains the Lagrangian structure of the Euler equations, as well as most of its conservation laws and local existence theory. Despite this, we give three different constructions of such an operator $A$ which admits smooth solutions that blow up in finite time, including an example on ${\bf R}^3$ which is self-adjoint and positive definite. This indicates a barrier to establishing global regularity for the three-dimensional Euler equations, in that any such method must use some property of those equations that is not shared by the generalised Euler equations considered here.

math.AP↗

arXiv2016

The logarithmically averaged Chowla and Elliott conjectures for two-point correlations

Let $λ$ denote the Liouville function. The Chowla conjecture, in the two-point correlation case, asserts that $$ \sum_{n \leq x} λ(a_1 n + b_1) λ(a_2 n+b_2) = o(x) $$ as $x \to \infty$, for any fixed natural numbers $a_1,a_2,b_1,b_2$ with $a_1b_2-a_2b_1 \neq 0$. In this paper we establish the logarithmically averaged version $$ \sum_{x/ω(x) < n \leq x} \frac{λ(a_1 n + b_1) λ(a_2 n+b_2)}{n} = o(\log ω(x)) $$ of the Chowla conjecture as $x \to \infty$, where $1 \leq ω(x) \leq x$ is an arbitrary function of $x$ that goes to infinity as $x \to \infty$, thus breaking the "parity barrier" for this problem. Our main tools are the multiplicativity of the Liouville function at small primes, a recent result of Matomäki, Radziwiłł, and the author on the averages of modulated multiplicative functions in short intervals, concentration of measure inequalities, the Hardy-Littlewood circle method combined with a restriction theorem for the primes, and a novel "entropy decrement argument". Most of these ingredients are also available (in principle, at least) for the higher order correlations, with the main missing ingredient being the need to control short sums of multiplicative functions modulated by local nilsequences. Our arguments also extend to more general bounded multiplicative functions than the Liouville function $λ$, leading to a logarithmically averaged version of the Elliott conjecture in the two-point case. In a subsequent paper we will use this version of the Elliott conjecture to affirmatively settle the Erdős discrepancy problem.

math.NT↗

arXiv2016

Long gaps between primes

Let $p_n$ denotes the $n$-th prime. We prove that $$\max_{p_{n+1} \leq X} (p_{n+1}-p_n) \gg \frac{\log X \log \log X\log\log\log\log X}{\log \log \log X}$$ for sufficiently large $X$, improving upon recent bounds of the first three and fifth authors and of the fourth author. Our main new ingredient is a generalization of a hypergraph covering theorem of Pippenger and Spencer, proven using the Rödl nibble method.

math.NT↗

arXiv2016

Equivalence of the logarithmically averaged Chowla and Sarnak conjectures

Let $λ$ denote the Liouville function. The Chowla conjecture asserts that $$ \sum_{n \leq X} λ(a_1 n + b_1) λ(a_2 n+b_2) \dots λ(a_k n + b_k) = o_{X \to \infty}(X) $$ for any fixed natural numbers $a_1,a_2,\dots,a_k$ and non-negative integer $b_1,b_2,\dots,b_k$ with $a_ib_j-a_jb_i \neq 0$ for all $1 \leq i < j \leq k$, and any $X \geq 1$. This conjecture is open for $k \geq 2$. As is well known, this conjecture implies the conjecture of Sarnak that $$ \sum_{n \leq X} λ(n) f(n) = o_{X \to \infty}(X)$$ whenever $f : {\bf N} \to {\bf C}$ is a fixed deterministic sequence and $X \geq 1$. In this paper, we consider the weaker logarithmically averaged versions of these conjectures, namely that $$ \sum_{X/ω\leq n \leq X} \frac{λ(a_1 n + b_1) λ(a_2 n+b_2) \dots λ(a_k n + b_k)}{n} = o_{ω\to \infty}(\log ω) $$ and $$ \sum_{X/ω\leq n \leq X} \frac{λ(n) f(n)}{n} = o_{ω\to \infty}(\log ω)$$ under the same hypotheses on $a_1,\dots,a_k,b_1,\dots,b_k$ and $f$, and for any $2 \leq ω\leq X$. Our main result is that these latter two conjectures are logically equivalent to each other, as well as to the "local Gowers uniformity" of the Liouville function. The main tools used here are the entropy decrement argument of the author used recently to establish the $k=2$ case of the logarithmically averaged Chowla conjecture, as well as the inverse conjecture for the Gowers norms, obtained by Green, Ziegler, and the author.

math.NT↗

arXiv2016

Finite time blowup for a supercritical defocusing nonlinear wave system

We consider the global regularity problem for defocusing nonlinear wave systems $$ \Box u = (\nabla_{{\bf R}^m} F)(u) $$ on Minkowski spacetime ${\bf R}^{1+d}$ with d'Alambertian $\Box := -\partial_t^2 + \sum_{i=1}^d \partial_{x_i}^2$, the field $u: {\bf R}^{1+d} \to {\bf R}^m$ is vector-valued, and $F: {\bf R}^m \to {\bf R}$ is a smooth potential which is positive and homogeneous of order $p+1$ outside of the unit ball, for some $p >1$. This generalises the scalar defocusing nonlinear wave (NLW) equation, in which $m=1$ and $F(v) = \frac{1}{p+1} |v|^{p+1}$. It is well known that in the energy sub-critical and energy-critical cases when $d \leq 2$ or $d \geq 3$ and $p \leq 1+\frac{4}{d-2}$, one has global existence of smooth solutions (for dimensions $d \leq 7$ at least) from arbitrary smooth initial data $u(0), \partial_t u(0)$. In this paper we study the supercritical case where $d = 3$ and $p > 5$. We show that in this case, there exists smooth potential $F$ for some sufficiently large $m$ (in fact we can take $m=40$), positive and homogeneous of order $p+1$ outside of the unit ball, and a smooth choice of initial data $u(0), \partial_t u(0)$ for which the solution develops a finite time singularity. In fact the solution is discretely self-similar in a backwards light cone. The basic strategy is to first select the mass and energy densities of $u$, then $u$ itself, and then finally design the potential $F$ in order to solve the required equation. The Nash embedding theorem is used in the second step, explaining the need to take $m$ relatively large.

math.AP↗

arXiv2016

Finite time blowup for high dimensional nonlinear wave systems with bounded smooth nonlinearity

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 function with $f(0)=0$ and all derivatives bounded; the higher-dimensional sine-Gordon equation $\Box u = \sin u$ is a model example of this class of nonlinear wave system. For dimensions $d \leq 9$, it follows from the work of Heinz, Pecher, Brenner, and von Wahl that one has smooth solutions to this equation for any smooth choice of initial data. Perhaps surprisingly, we show that this result is almost sharp, in the sense that for any $d \geq 11$, there exists an $m$ (in fact we can take $m=2$) and a nonlinearity $f \colon {\bf R}^m \to {\bf R}^m$ with all derivatives bounded, for which the above equation admits solutions that blow up in finite time. The intermediate case $d=10$ remains open.

math.AP↗

arXiv2016

Polynomial patterns in the primes

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 numbers $n,m$ such that $n+P_1(m),\dots,n+P_k(m)$ are simultaneously prime, generalizing a previous result of the authors, which was restricted to the special case $P_1(0)=\dots=P_k(0)=0$ (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 $n \leq N$ and $m \leq M$ with $M$ slightly less than $N^{1/d}$. 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 $P_1(0)=\dots=P_k(0)=0$, our methods also give infinitely many $n,m$ with $n+P_1(m),\dots,n+P_k(m)$ 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 $δ$.

math.NT↗

arXiv2016

Sumfree sets in groups: a survey

We discuss several questions concerning sum-free sets in groups, raised by Erdős 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 that the sum of any two different elements of $B$ do not belong to $A$ (in other words, $B$ is sum-free with respect to $A$). Erdős, in the above mentioned survey, conjectured that if $|A|$ is sufficiently large compared to $k$, then $A$ contains two elements that add up to zero. This is known to be true for $k \leq 3$. We give counterexamples for all $k \ge 4$. On the other hand, using the new characterization result, we are able to prove a positive result in the case when $|G|$ is not divisible by small primes.

math.CO↗

arXiv2015

Chains of large gaps between primes

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 \[ G_1(X) \gg \frac{\log X \log \log X\log\log\log\log X}{\log \log \log X}\] for sufficiently large $X$. In this note, we combine the arguments in that paper with the Maier matrix method to show that \[ G_k(X) \gg \frac{1}{k^2} \frac{\log X \log \log X\log\log\log\log X}{\log \log \log X}\] for any fixed $k$ and sufficiently large $X$. The implied constant is effective and independent of $k$.

math.NT↗

arXiv2015

Large gaps between consecutive prime numbers

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 set of primes by arithmetic progressions. As such, we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes.

math.NT↗

arXiv2015

Inverse theorems for sets and measures of polynomial growth

We give a structural description of the finite subsets $A$ of an arbitrary group $G$ which obey the polynomial growth condition $|A^n| \leq n^d |A|$ for some bounded $d$ and sufficiently large $n$, showing that such sets are controlled by (a bounded number of translates of) a coset nilprogression in a certain precise sense. This description recovers some previous results of Breuillard-Green-Tao and Breuillard-Tointon concerning sets of polynomial growth; we are also able to describe the subsequent growth of $|A^m|$ fairly explicitly for $m \geq n$, at least when $A$ is a symmetric neighbourhood of the identity. We also obtain an analogous description of symmetric probability measures $μ$ whose $n$-fold convolutions $μ^{*n}$ obey the condition $\| μ^{*n} \|_{\ell^2}^{-2} \leq n^d \|μ\|_{\ell^2}^{-2}$. In the abelian case, this description recovers the inverse Littlewood-Offord theorem of Nguyen-Vu, and gives a variant of a recent nonabelian inverse Littlewood-Offord theorem of Tiep-Vu. Our main tool to establish these results is the inverse theorem of Breuillard, Green, and the author that describes the structure of approximate groups.

math.CO↗

arXiv2015

Sign patterns of the Liouville and Möbius functions

Let $λ$ and $μ$ denote the Liouville and Möbius functions respectively. Hildebrand showed that all eight possible sign patterns for $(λ(n), λ(n+1), λ(n+2))$ 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 result by proving that each of these eight sign patterns occur with positive lower natural density. We also obtain an analogous result for the nine possible sign patterns for $(μ(n), μ(n+1))$. A new feature in the latter argument is the need to demonstrate that a certain random graph is almost surely connected.

math.NT↗

arXiv2015

The quantitative behaviour of polynomial orbits on nilmanifolds

A theorem of Leibman asserts that a polynomial orbit $(g(1),g(2),g(3),\ldots)$ on a nilmanifold $G/Γ$ is always equidistributed in a union of closed sub-nilmanifolds 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 $(g(1),\ldots,g(N))$ in a nilmanifold. More specifically we show that there is a factorization $g = εg'γ$, where $ε(n)$ is "smooth", $γ(n)$ is periodic and "rational", and $(g'(a),g'(a+d),\ldots,g'(a + d(l-1)))$ is uniformly distributed (up to a specified error $δ$) inside some subnilmanifold $G'/Γ'$ of $G/Γ$, for all sufficiently dense arithmetic progressions $a,a+d,\ldots,a+d(l-1)$ inside $\{1,..,N\}$. Our bounds are uniform in $N$ and are polynomial in the error tolerance delta. In a subsequent paper we shall use this theorem to establish the Mobius and Nilsequences conjecture from our earlier paper "Linear equations in primes".

math.NT↗

arXiv2015

On the quantitative distribution of polynomial nilsequences - erratum

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, Theorem 8.6] from the 1-parameter results in [GT]. This is the same basic mode of argument we attempted in the original paper, though the details are different. The equal sides case is the only one required in applications such as the proof of the inverse conjectures for the Gowers norms due to the authors and Ziegler. Second, we sketch a proof that [GT, Theorem 8.6] does in fact hold in its originally stated form, that is to say without the equal sides condition. To obtain this statement the entire argument of [GT] must be run in the context of multiparameter polynomial sequences $g : \mathbb{Z}^t \rightarrow G$ rather than 1-parameter sequences $g : \mathbb{Z} \rightarrow G$ as is currently done.

math.NT↗

arXiv2015

Counting the number of solutions to the Erdos-Straus equation on unit fractions

For any positive integer $n$, let $f(n)$ denote the number of solutions to the Diophantine equation $\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ with $x,y,z$ positive integers. The \emph{Erdős-Straus conjecture} asserts that $f(n) > 0$ for every $n \geq 2$. To solve this conjecture, it suffices without loss of generality to consider the case when $n$ is a prime $p$. In this paper we consider the question of bounding the sum $\sum_{p<N} f(p)$ asymptotically as $N \to \infty$, where $p$ ranges over primes. Our main result establishes the asymptotic upper and lower bounds $$ N \log^2 N \ll \sum_{p \leq N} f(p) \ll N \log^2 N \log \log N.$$ In particular, from this bound and the prime number theorem we have $f(p) = O(\log^3 p \log \log p)$ for a subset of primes of density arbitrarily close to 1; thus a typical prime has a relatively small number of solutions to the Erdős-Straus Diophantine equation. We also establish some related results on $f$ and related quantities, for instance establishing the bound $f(p) \ll p^{3/5} + O(\frac{1}{\log\log p})}$ for all primes $p$.

math.NT↗

arXiv2015

Cancellation for the multilinear Hilbert transform

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 $1 < p_1,\dots,p_k,p < \infty$ and $\frac{1}{p} = \frac{1}{p_1} + \dots + \frac{1}{p_k}$. This is proven for $k=1,2$, but remains open for larger $k$. In this paper, we consider the truncated operators $$ H_{k,r,R}( f_1,\dots,f_k )(x) := \int_{r \leq |t| \leq R} f_1(x+t) \dots f_k(x+kt)\ \frac{dt}{t}$$ for $R > r > 0$. The above conjecture is equivalent to the uniform boundedness of $\| H_{k,r,R} \|_{L^{p_1}({\bf R}) \times \dots \times L^{p_k}({\bf R}) \to L^p({\bf R})}$ in $r,R$, whereas the Minkowski and Hölder inequalities give the trivial upper bound of $2 \log \frac{R}{r}$ for this quantity. By using the arithmetic regularity and counting lemmas of Green and the author, we improve the trivial upper bound on $\| H_{k,r,R} \|_{L^{p_1}({\bf R}) \times \dots \times L^{p_k}({\bf R}) \to L^p({\bf R})}$ slightly to $o( \log \frac{R}{r} )$ in the limit $\frac{R}{r} \to \infty$ for any admissible choice of $k$ and $p_1,\dots,p_k,p$. This establishes some cancellation in the $k$-linear Hilbert transform $H_k$, but not enough to establish its boundedness in $L^p$ spaces.

math.CA↗

arXiv2015

Failure of the $L^1$ pointwise and maximal ergodic theorems for the free group

Let $F_2$ denote the free group on two generators $a,b$. For any measure-preserving system $(X, {\mathcal X}, μ, (T_g)_{g \in F_2})$ on a finite measure space $X = (X,{\mathcal X},μ)$, any $f \in L^1(X)$, and any $n \geq 1$, define the averaging operators $${\mathcal A}_n f(x) := \frac{1}{4 \times 3^{n-1}} \sum_{g \in F_2: |g| = n} f( T_g^{-1} x ),$$ where $|g|$ denotes the word length of $g$. We give an example of a measure-preserving system $X$ and an $f \in L^1(X)$ such that the sequence ${\mathcal A}_n f(x)$ is unbounded in $n$ for almost every $x$, thus showing that the pointwise and maximal ergodic theorems do not hold in $L^1$ for actions of $F_2$. This is despite the results of Nevo-Stein and Bufetov, who establish pointwise and maximal ergodic theorems in $L^p$ for $p>1$ and for $L \log L$ respectively, as well as an estimate of Naor and the author establishing a weak-type $(1,1)$ maximal inequality for the action on $\ell^1(F_2)$. Our construction is a variant of a counterexample of Ornstein concerning iterates of a Markov operator.

math.DS↗