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Article
Infinite partial sumsets in the primes
We show that there exist infinite sets A = (a1, a2, …} and B = {b1, b2, …} of natural numbers such that ai + bj is prime whenever 1 ≤ i < j.
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Article
Multiple recurrence and convergence results associated to \({\text{F}}_P^\omega \) -actions
Using an ergodic inverse theorem obtained in our previous paper, we obtain limit formulae for multiple ergodic averages associated with the action of ...
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Article
Large values of the Gowers-Host-Kra seminorms
The Gowers uniformity norms \({\left\| f \right\|_{{U^k}(G)}}\) of a function f: G → C on a finite ad...
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Article
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 determini...
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Article
The Gaussian primes contain arbitrarily shaped constellations
We show that the Gaussian primesP[i] ⊆ ℤ[i] contain infinitely constellations of any prescribed shape and orientation. More precisely, we show that given any distinct Gaussian integersv 0,...
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Article
Uniform estimates on paraproducts
We prove uniformL p estimates (Theorem 1.1) for a family of paraproducts and corresponding maximal operators.
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Article
A physical space approach to wave equation bilinear estimates
Bilinear estimates for the wave equation in Minkowski space are normally proven using the Fourier transform and Plancherel’s theorem. However, such methods are difficult to carry over to non-flat situations (s...
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Article
Uniform estimates on multi-linear operators with modulation symmetry
In a previous paper [20] in this series, we gaveL p estimates for multi-linear operators given by multipliers which are singular on a non-degenerate subspace of some dimensionk. In this pa...
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Article
New bounds for Kakeya problems
We establish new estimates on the Minkowski and Hausdorff dimensions of Kakeya sets and we obtain new bounds on the Kakeya maximal operator.