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Arithmetic progressions and the primes.

Terence Tao — 2006

Collectanea Mathematica

We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length k in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes.

Geometric renormalization of large energy wave maps

Terence Tao — 2004

Journées Équations aux dérivées partielles

There has been much progress in recent years in understanding the existence problem for wave maps with small critical Sobolev norm (in particular for two-dimensional wave maps with small energy); a key aspect in that theory has been a renormalization procedure (either a geometric Coulomb gauge, or a microlocal gauge) which converts the nonlinear term into one closer to that of a semilinear wave equation. However, both of these renormalization procedures encounter difficulty if the energy of the...

Restriction theory of the Selberg sieve, with applications

Ben GreenTerence Tao — 2006

Journal de Théorie des Nombres de Bordeaux

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. Write h ( θ ) : = n X e ( n θ ) , where X is the set of all n N such that the numbers a 1 n + b 1 , , a k n + b k are all prime. We obtain upper bounds for h L p ( 𝕋 ) , p > 2 , which are (conditionally on the Hardy-Littlewood prime tuple conjecture) of the correct order...

Carleson measures, trees, extrapolation, and T(b) theorems.

Pascal AuscherSteve HofmannCamil MuscaluTerence TaoChristoph Thiele — 2002

Publicacions Matemàtiques

The theory of Carleson measures, stopping time arguments, and atomic decompositions has been well-established in harmonic analysis. More recent is the theory of phase space analysis from the point of view of wave packets on tiles, tree selection algorithms, and tree size estimates. The purpose of this paper is to demonstrate that the two theories are in fact closely related, by taking existing results and reproving them in a unified setting. In particular we give a dyadic version of extrapolation...

A variation norm Carleson theorem

Richard OberlinAndreas SeegerTerence TaoChristoph ThieleJames Wright — 2012

Journal of the European Mathematical Society

We strengthen the Carleson-Hunt theorem by proving L p estimates for the r -variation of the partial sum operators for Fourier series and integrals, for r > 𝚖𝚊𝚡 { p ' , 2 } . Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.

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