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.
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...
The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an – restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime -tuples. Let and be positive integers. Write , where is the set of all such that the numbers are all prime. We obtain upper bounds for , , which are (conditionally on the Hardy-Littlewood prime tuple conjecture) of the correct order...
We establish sharp (H,L) and local (L logL,L) mapping properties for rough one-dimensional multipliers. In particular, we show that the multipliers in the Marcinkiewicz multiplier theorem map H to L and L logL to L, and that these estimates are sharp.
We prove an x-ray estimate in general dimension which is a stronger version of Wolff's Kakeya estimate [12]. This generalizes the estimate in [13], which dealt with the n = 3 case.
We survey recent developments on the Kakeya problem. [Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial (Madrid), 2002].
We prove the “Möbius and Nilsequences Conjecture” for nilsystems of step 1 and 2. This paper forms a part of our program to generalise the Hardy-Littlewood method so as to handle of linear equations in primes.
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...
We prove that classical Coifman-Meyer theorem holds on any polidisc T or arbitrary dimension d ≥ 1.
We strengthen the Carleson-Hunt theorem by proving estimates for the -variation of the partial sum operators for Fourier series and integrals, for . Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.
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 [BGGT].
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