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Terence Tao

Publications and source records attributed to Terence Tao.

At least 253 records · Page 14Linked to original sources
arXiv2007

Norm convergence of multiple ergodic averages for commuting transformations

Let $T_1, ..., T_l: X \to X$ be commuting measure-preserving transformations on a probability space $(X, \X, μ)$. We show that the multiple ergodic averages $\frac{1}{N} \sum_{n=0}^{N-1} f_1(T_1^n x) ... f_l(T_l^n x)$ are convergent in $L^2(X,\X,μ)$ as $N \to \infty$ for all $f_1,...,f_l \in L^\infty(X,\X,μ)$; this was previously established for $l=2$ by Conze and Lesigne and for general $l$ assuming some additional ergodicity hypotheses on the maps $T_i$ and $T_i T_j^{-1}$ by Frantzikinakis and Kra (with the $l=3$ case of this result established earlier by Zhang). Our approach is combinatorial and finitary in nature, inspired by recent developments regarding the hypergraph regularity and removal lemmas, although we will not need the full strength of those lemmas. In particular, the $l=2$ case of our arguments are a finitary analogue of those of Conze and Lesigne.

math.DS↗
arXiv2007

The primes contain arbitrarily long arithmetic progressions

We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi's theorem, which asserts that any subset of the integers of positive density contains progressions of arbitrary length. The second, which is the main new ingredient of this paper, is a certain transference principle. This allows us to deduce from Szemeredi's theorem that any subset of a sufficiently pseudorandom set of positive relative density contains progressions of arbitrary length. The third ingredient is a recent result of Goldston and Yildirim. Using this, one may place the primes inside a pseudorandom set of ``almost primes'' with positive relative density.

math.NT↗
arXiv2007

Quadratic Uniformity of the Mobius Function

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 [LEP], which accompanies this submission. In particular, the results of this paper may be used, together with the machinery of [LEP], 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 Mobius and Nilsequences Conjecture for 1 and 2-step nilsequences. This conjecture is introduced in [LEP] and amounts to showing that if G/Γis an s-step nilmanifold, s <= 2, if F : G/Γ-> [-1,1] is a Lipschitz function, and if T_g : G/Γ-> G/Γis the action of g \in G on G/Γ, then the Mobius function μ(n) is orthogonal to the sequence F(T_g^n x) in a fairly strong sense, uniformly in g and x in G/Γ. This can be viewed as a ``quadratic'' generalisation of an exponential sum estimate of Davenport, and is proven by the following the methods of Vinogradov and Vaughan.

math.NT↗
arXiv2007

Breaking the duality in the return times theorem

We prove Bourgain's Return Times Theorem for a range of exponents $p$ and $q$ that are outside the duality range. An oscillation result is used to prove hitherto unknown almost everywhere convergence for the signed average analog of Bourgain's averages. As an immediate corollary we obtain a Wiener-Wintner type of result for the ergodic Hilbert series.

math.DS↗
arXiv2007

Structure and randomness in combinatorics

Combinatorics, like computer science, often has to deal with large objects of unspecified (or unusable) structure. One powerful way to deal with such an arbitrary object is to decompose it into more usable components. In particular, it has proven profitable to decompose such objects into a \emph{structured} component, a \emph{pseudo-random} component, and a \emph{small} component (i.e. an error term); in many cases it is the structured component which then dominates. We illustrate this philosophy in a number of model cases.

math.CO↗
arXiv2007

A correspondence principle between (hyper)graph theory and probability theory, and the (hyper)graph removal lemma

We introduce a correspondence principle (analogous to the Furstenberg correspondence principle) that allows one to extract an infinite random graph or hypergraph from a sequence of increasingly large deterministic graphs or hypergraphs. As an application we present a new (infinitary) proof of the hypergraph removal lemma of Nagle-Schacht-Rödl-Skokan and Gowers, which does not require the hypergraph regularity lemma and requires significantly less computation. This in turn gives new proofs of several corollaries of the hypergraph removal lemma, such as Szemerédi's theorem on arithmetic progressions.

math.CO↗
arXiv2007

Freiman's theorem in finite fields via extremal set theory

Using various results from extremal set theory (interpreted in the language of additive combinatorics), we prove an asyptotically sharp version of Freiman's theorem in F_2^n: if A in F_2^n is a set for which |A + A| <= K|A| then A is contained in a subspace of size 2^{2K + O(\sqrt{K}\log K)}|A|; except for the O(\sqrt{K} \log K) error, this is best possible. If in addition we assume that A is a downset, then we can also cover A by O(K^{46}) translates of a coordinate subspace of size at most |A|, thereby verifying the so-called polynomial Freiman-Ruzsa conjecture in this case. A common theme in the arguments is the use of compression techniques. These have long been familiar in extremal set theory, but have been used only rarely in the additive combinatorics literature.

math.CO↗
arXiv2007

The condition number of a randomly perturbed matrix

Let $M$ be an arbitrary $n$ by $n$ matrix. We study the condition number a random perturbation $M+N_n$ of $M$, where $N_n$ is a random matrix. It is shown that, under very general conditions on $M$ and $M_n$, the condition number of $M+N_n$ is polynomial in $n$ with very high probability. The main novelty here is that we allow $N_n$ to have discrete distribution.

math.PR↗
arXiv2007

What is good mathematics?

Some personal thoughts and opinions on what ``good quality mathematics'' is, and whether one should try to define this term rigorously. As a case study, the story of Szemerédi's theorem is presented.

math.HO↗
arXiv2007

Inverse Littlewood-Offord theorems and the condition number of random discrete matrices

Consider a random sum $η_1 v_1 + ... + η_n v_n$, where $η_1,...,η_n$ are i.i.d. random signs and $v_1,...,v_n$ are integers. The Littlewood-Offord problem asks to maximize concentration probabilities such as $¶(η_1 v_1 + ... + η_n v_n = 0)$ subject to various hypotheses on the $v_1,...,v_n$. In this paper we develop an \emph{inverse} Littlewood-Offord theorem (somewhat in the spirit of Freiman's inverse sumset theorem), which starts with the hypothesis that a concentration probability is large, and concludes that almost all of the $v_1,...,v_n$ are efficiently contained in an arithmetic progression. As an application we give some new bounds on the distribution of the least singular value of a random Bernoulli matrix, which in turn gives upper tail estimates on the condition number.

math.PR↗
arXiv2006

A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order

Solutions to the Cauchy problem for the one-dimensional cubic nonlinear Schrödinger equation on the real line are studied in Sobolev spaces $H^s$, for $s$ negative but close to 0. For smooth solutions there is an {\em a priori} upper bound for the $H^s$ norm of the solution, in terms of the $H^s$ norm of the datum, for arbitrarily large data, for sufficiently short time. Weak solutions are constructed for arbitrary initial data in $H^s$.

math.AP↗
arXiv2006

Minimal-mass blowup solutions of the mass-critical NLS

We consider the minimal mass $m_0$ required for solutions to the mass-critical nonlinear Schrödinger (NLS) equation $iu_t + Δu = μ|u|^{4/d} u$ to blow up. If $m_0$ is finite, we show that there exists a minimal-mass solution blowing up (in the sense of an infinite spacetime norm) in both time directions, whose orbit in $L^2_x(\R^d)$ is compact after quotienting out by the symmetries of the equation. A similar result is obtained for spherically symmetric solutions. Similar results were previously obtained by Keraani, \cite{keraani}, in dimensions 1, 2 and Begout and Vargas, \cite{begout}, in dimensions $d\geq 3$ for the mass-critical NLS and by Kenig and Merle, \cite{merlekenig}, in the energy-critical case. In a subsequent paper we shall use this compactness result to establish global existence and scattering in $L^2_x(\R^d)$ for the defocusing NLS in three and higher dimensions with spherically symmetric data.

math.AP↗
arXiv2006

Global well-posedness and scattering for the mass-critical nonlinear Schrödinger equation for radial data in high dimensions

We establish global well-posedness and scattering for solutions to the defocusing mass-critical (pseudoconformal) nonlinear Schrödinger equation $iu_t + Δu = |u|^{4/n} u$ for large spherically symmetric $L^2_x(\R^n)$ initial data in dimensions $n\geq 3$. After using the reductions in \cite{compact} to reduce to eliminating blowup solutions which are almost periodic modulo scaling, we obtain a frequency-localized Morawetz estimate and exclude a mass evacuation scenario (somewhat analogously to \cite{ckstt:gwp}, \cite{RV}, \cite{thesis:art}) in order to conclude the argument.

math.AP↗
arXiv2006

A counterexample to an endpoint bilinear Strichartz inequality

The endpoint Strichartz estimate $\| e^{itΔ} f \|_{L^2_t L^\infty_x(\R \times \R^2)} \lesssim \|f\|_{L^2_x(\R^2)}$ is known to be false by the work of Montgomery-Smith, despite being only ``logarithmically far'' from being true in some sense. In this short note we show that (in sharp constrast to the $L^p_{t,x}$ Strichartz estimates) the situation is not improved by passing to a bilinear setting; more precisely, if $P, P'$ are non-trivial smooth Fourier cutoff multipliers then we show that the bilinear estimate $$\| (e^{itΔ} P f) (e^{itΔ} P' g) \|_{L^2_t L^\infty_x(\R \times \R^2)} \lesssim \|f\|_{L^2_x(\R^2)} \|g\|_{L^2_x(\R^2)} $$ fails even when $P$, $P'$ have widely separated supports.

math.AP↗
arXiv2006

Global behaviour of nonlinear dispersive and wave equations

We survey recent advances in the analysis of the large data global (and asymptotic) behaviour of nonlinear dispersive equations such as the nonlinear wave (NLW), nonlinear Schrödinger (NLS), wave maps (WM), Schrödinger maps (SM), generalised Korteweg-de Vries (gKdV), Maxwell-Klein-Gordon (MKG), and Yang-Mills (YM) equations. The classification of the nonlinearity as \emph{subcritical} (weaker than the linear dispersion at high frequencies), \emph{critical} (comparable to the linear dispersion at all frequencies), or \emph{supercritical} (stronger than the linear dispersion at high frequencies) is fundamental to this analysis, and much of the recent progress has pivoted on the case when there is a critical conservation law. We discuss how one synthesises a satisfactory critical (scale-invariant) global theory, starting the basic building blocks of perturbative analysis, conservation laws, and monotonicity formulae, but also incorporating more advanced (and recent) tools such as gauge transforms, concentration-compactness, and induction on energy.

math.AP↗
arXiv2006

Multilinear estimates for periodic KdV equations and applications

We prove an endpoint multilinear estimate for the $X^{s,b}$ spaces associated to the periodic Airy equation. As a consequence we obtain sharp local well-posedness results for periodic generalized KdV equations, as well as some global well-posedness results below the energy norm. In particular we prove a multilinear estimate which completes the proof of global well-posedness for periodic KdV in a preceding paper (math.AP/0110045) down to the optimal regularity H^{-1/2}.

math.AP↗
arXiv2006

Global regularity for a logarithmically supercritical defocusing nonlinear wave equation for spherically symmetric data

We establish global regularity for the logarithmically energy-supercritical wave equation $\Box u = u^5 \log(2+u^2)$ in three spatial dimensions for spherically symmetric initial data, by modifying an argument of Ginibre, Soffer and Velo \cite{gsv} for the energy-critical equation. This example demonstrates that critical regularity arguments can penetrate very slightly into the supercritical regime.

math.AP↗