arXiv subjects

Terence Tao

Publications and source records attributed to Terence Tao.

At least 271 records · Page 15Linked to original sources

arXiv2006

Scattering for the quartic generalised Korteweg-de Vries equation

We show that the quartic generalised KdV equation $$ u_t + u_{xxx} + (u^4)_x = 0$$ is globally wellposed for data in the critical (scale-invariant) space $\dot H^{-1/6}_x(\R)$ with small norm (and locally wellposed for large norm), improving a result of Grünrock. As an application we obtain scattering results in $H^1_x(\R) \cap \dot H^{-1/6}_x(\R)$ for the radiation component of a perturbed soliton for this equation, improving the asymptotic stability results of Martel and Merle.

math.AP↗

arXiv2006

The ergodic and combinatorial approaches to Szemerédi's theorem

A famous theorem of Szemerédi asserts that any set of integers of positive upper density will contain arbitrarily long arithmetic progressions. In its full generality, we know of four types of arguments that can prove this theorem: the original combinatorial (and graph-theoretical) approach of Szemerédi, the ergodic theory approach of Furstenberg, the Fourier-analytic approach of Gowers, and the hypergraph approach of Nagle-Rödl-Schacht-Skokan and Gowers. In this lecture series we introduce the first, second and fourth approaches, though we will not delve into the full details of any of them. One of the themes of these lectures is the strong similarity of ideas between these approaches, despite the fact that they initially seem rather different.

math.CO↗

arXiv2006

Near Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?

Suppose we are given a vector $f$ in $\R^N$. How many linear measurements do we need to make about $f$ to be able to recover $f$ to within precision $ε$ in the Euclidean ($\ell_2$) metric? Or more exactly, suppose we are interested in a class ${\cal F}$ of such objects--discrete digital signals, images, etc; how many linear measurements do we need to recover objects from this class to within accuracy $ε$? This paper shows that if the objects of interest are sparse or compressible in the sense that the reordered entries of a signal $f \in {\cal F}$ decay like a power-law (or if the coefficient sequence of $f$ in a fixed basis decays like a power-law), then it is possible to reconstruct $f$ to within very high accuracy from a small number of random measurements.

math.CA↗

arXiv2006

A sum-product estimate in finite fields, and applications

Let $A$ be a subset of a finite field $F := \Z/q\Z$ for some prime $q$. If $|F|^δ< |A| < |F|^{1-δ}$ for some $δ> 0$, then we prove the estimate $|A+A| + |A.A| \geq c(δ) |A|^{1+\eps}$ for some $\eps = \eps(δ) > 0$. This is a finite field analogue of a result of Erdos and Szemeredi. We then use this estimate to prove a Szemeredi-Trotter type theorem in finite fields, and obtain a new estimate for the Erdos distance problem in finite fields, as well as the three-dimensional Kakeya problem in finite fields.

math.CO↗

arXiv2006

Compressions, convex geometry and the Freiman-Bilu theorem

We note a link between combinatorial results of Bollobás and Leader concerning sumsets in the grid, the Brunn-Minkowski theorem and a result of Freiman and Bilu concerning the structure of sets of integers with small doubling. Our main result is the following. If eps > 0 and if A is a finite nonempty subset of a torsion-free abelian group with |A + A| <= K|A|, then A may be covered by exp(K^C) progressions of dimension [log_2 K + eps] and size at most |A|.

math.NT↗

arXiv2005

The dichotomy between structure and randomness, arithmetic progressions, and the primes

A famous theorem of Szemerédi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness, which in turn leads (roughly speaking) to a decomposition of any object into a structured (low-complexity) component and a random (discorrelated) component. Important examples of these types of decompositions include the Furstenberg structure theorem and the Szemerédi regularity lemma. One recent application of this dichotomy is the result of Green and Tao establishing that the prime numbers contain arbitrarily long arithmetic progressions (despite having density zero in the integers). The power of this dichotomy is evidenced by the fact that the Green-Tao theorem requires surprisingly little technology from analytic number theory, relying instead almost exclusively on manifestations of this dichotomy such as Szemerédi's theorem. In this paper we survey various manifestations of this dichotomy in combinatorics, harmonic analysis, ergodic theory, and number theory. As we hope to emphasize here, the underlying themes in these arguments are remarkably similar even though the contexts are radically different.

math.NT↗

arXiv2005

Stable Signal Recovery from Incomplete and Inaccurate Measurements

Suppose we wish to recover an n-dimensional real-valued vector x_0 (e.g. a digital signal or image) from incomplete and contaminated observations y = A x_0 + e; A is a n by m matrix with far fewer rows than columns (n << m) and e is an error term. Is it possible to recover x_0 accurately based on the data y? To recover x_0, we consider the solution x* to the l1-regularization problem min \|x\|_1 subject to \|Ax-y\|_2 <= epsilon, where epsilon is the size of the error term e. We show that if A obeys a uniform uncertainty principle (with unit-normed columns) and if the vector x_0 is sufficiently sparse, then the solution is within the noise level \|x* - x_0\|_2 \le C epsilon. As a first example, suppose that A is a Gaussian random matrix, then stable recovery occurs for almost all such A's provided that the number of nonzeros of x_0 is of about the same order as the number of observations. Second, suppose one observes few Fourier samples of x_0, then stable recovery occurs for almost any set of p coefficients provided that the number of nonzeros is of the order of n/[\log m]^6. In the case where the error term vanishes, the recovery is of course exact, and this work actually provides novel insights on the exact recovery phenomenon discussed in earlier papers. The methodology also explains why one can also very nearly recover approximately sparse signals.

math.NA↗

arXiv2005

A variant of the hypergraph removal lemma

Recent work of Gowers and Nagle, Rödl, Schacht, and Skokan has established a hypergraph removal lemma, which in turn implies some results of Szemerédi and Furstenberg-Katznelson concerning one-dimensional and multi-dimensional arithmetic progressions respectively. In this paper we shall give a self-contained proof of this hypergraph removal lemma. In fact we prove a slight strengthening of the result, which we will use in a subsequent paper to establish infinitely many constellations of a prescribed shape in the Gaussian primes.

math.CO↗

arXiv2005

Szemerédi's regularity lemma revisited

Szemerédi's regularity lemma is a basic tool in graph theory, and also plays an important role in additive combinatorics, most notably in proving Szemerédi's theorem on arithmetic progressions . In this note we revisit this lemma from the perspective of probability theory and information theory instead of graph theory, and observe a variant of this lemma which introduces a new parameter $F$. This stronger version of the regularity lemma was iterated in a recent paper of the author to reprove the analogous regularity lemma for hypergraphs.

math.CO↗

arXiv2005

The Brascamp-Lieb inequalities: finiteness, structure, and extremals

We consider the Brascamp--Lieb inequalities concerning multilinear integrals of products of functions in several dimensions. We give a complete treatment of the issues of finiteness of the constant, and of the existence and uniqueness of centred gaussian extremals. For arbitrary extremals we completely address the issue of existence, and partly address the issue of uniqueness. We also analyse the inequalities from a structural perspective. We obtain two new proofs of Lieb's fundamental theorem concerning exhaustion by gaussians. Our techniques are partly based upon monotonicity formulas for positive solutions to heat equations in linear and multilinear settings.

math.MG↗

arXiv2005

The nonlinear Schrödinger equation with combined power-type nonlinearities

We undertake a comprehensive study of the nonlinear Schrödinger equation $$ i u_t +Δu = λ_1|u|^{p_1} u+ λ_2 |u|^{p_2} u, $$ where $u(t,x)$ is a complex-valued function in spacetime $\R_t\times\R^n_x$, $λ_1$ and $λ_2$ are nonzero real constants, and $0 0$ and $p_1=\frac{4}{n}$, $p_2=\frac{4}{n-2}$. The results at the endpoint $p_1 = \frac{4}{n}$ are conditional on a conjectured global existence and spacetime estimate for the $L^2_x$-critical nonlinear Schrödinger equation. As an off-shoot of our analysis, we also obtain a new, simpler proof of scattering in $H^1_x$ for solutions to the nonlinear Schrödinger equation $$ i u_t +Δu = |u|^{p} u, $$ with $\frac{4}{n}<p<\frac{4}{n-2}$, which was first obtained by J. Ginibre and G. Velo, \cite{gv:scatter}.

math.AP↗

arXiv2005

The Averaging lemma and regularizing effect

We prove new velocity averaging results for second-order multidimensional equations of the general form, $\op(\nabla_x,v)f(x,v)=g(x,v)$ where $\op(\nabla_x,v):=\bba(v)\cdot\nabla_x-\nabla_x^\top\cdot\bbb(v)\nabla_x$. These results quantify the Sobolev regularity of the averages, $\int_vf(x,v)ϕ(v)dv$, in terms of the non-degeneracy of the set $\{v: |\op(\ixi,v)|\leq δ\}$ and the mere integrability of the data, $(f,g)\in (L^p_{x,v},L^q_{x,v})$. Velocity averaging is then used to study the \emph{regularizing effect} in quasilinear second-order equations, $\op(\nabla_x,ρ)ρ=S(ρ)$ using their underlying kinetic formulations, $\op(\nabla_x,v)χ_ρ=g_{{}_S}$. In particular, we improve previous regularity statements for nonlinear conservation laws, and we derive completely new regularity results for convection-diffusion and elliptic equations driven by degenerate, non-isotropic diffusion.

math.AP↗

arXiv2005

On the Multilinear Restriction and Kakeya conjectures

We prove $d$-linear analogues of the classical restriction and Kakeya conjectures in $\R^d$. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians, closely related to heat flow. We conclude by giving some applications to the corresponding variable-coefficient problems and the so-called "joints" problem, as well as presenting some $n$-linear analogues for $n<d$.

math.CA↗

arXiv2005

Stability of energy-critical nonlinear Schrödinger equations in high dimensions

We develop the existence, uniqueness, continuity, stability, and scattering theory for energy-critical nonlinear Schrödinger equations in dimensions $n \geq 3$, for solutions which have large, but finite, energy and large, but finite, Strichartz norms. For dimensions $n \leq 6$, this theory is a standard extension of the small data well-posedness theory based on iteration in Strichartz spaces. However, in dimensions $n > 6$ there is an obstruction to this approach because of the subquadratic nature of the nonlinearity (which makes the derivative of the nonlinearity non-Lipschitz). We resolve this by iterating in exotic Strichartz spaces instead. The theory developed here will be applied in a subsequent paper of the second author, to establish global well-posedness and scattering for the defocusing energy-critical equation for large energy data.

math.AP↗

arXiv2005

Finite bounds for Holder-Brascamp-Lieb multilinear inequalities

A criterion is established for the validity of multilinear inequalities of a class considered by Brascamp and Lieb, generalizing well-known inequalities of Holder, Young, and Loomis-Whitney. This is a companion to a recent paper by the same authors, containing a different proof and discussing certain variants.

math.CA↗

arXiv2005

Obstructions to uniformity, and arithmetic patterns in the primes

In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing 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.

math.NT↗

arXiv2005

Restriction theory of the Selberg sieve, with applications

The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an L^2-L^p restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime k-tuples. Let a_1,...,a_k and b_1,...,b_k be positive integers. For t on the unit circle write h(t) := \sum_{n \in X} e(nt)$, where X is the set of all n <= N such that the numbers a_1n + b_1,..., a_kn + b_k are all prime. We obtain upper bounds for the L^p norm of h, p > 2, which are (conditionally on the prime tuple conjecture) of the correct order of magnitude. As a second application we deduce from Chen's theorem, Roth's theorem, and a transference principle that there are infinitely many arithmetic progressions p_1 < p_2 < p_3 of primes, such that p_i + 2 is either a prime or a product of two primes for each i=1,2,3.

math.NT↗

arXiv2005

Random symmetric matrices are almost surely non-singular

Let $Q_n$ denote a random symmetric $n$ by $n$ matrix, whose upper diagonal entries are i.i.d. Bernoulli random variables (which take values 0 and 1 with probability 1/2). We prove that $Q_n$ is non-singular with probability $1-O(n^{-1/8+δ})$ for any fixed $δ> 0$. The proof uses a quadratic version of Littlewood-Offord type results concerning the concentration functions of random variables and can be extended for more general models of random matrices.

math.PR↗