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Mixing for progressions in non-abelian groups

I’ve just uploaded to the arXiv my paper “Mixing for progressions in non-abelian groups“, submitted to Forum of Mathematics, Sigma (which, along with sister publication Forum of Mathematics, Pi, has...

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Small doubling in groups

Emmanuel Breuillard, Ben Green, and I have just uploaded to the arXiv our survey “Small doubling in groups“, for the proceedings of the upcoming Erdos Centennial.  This is a short survey of the known...

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A Fourier-free proof of the Furstenberg-Sarkozy theorem

The following result is due independently to Furstenberg and to Sarkozy: Theorem 1 (Furstenberg-Sarkozy theorem) Let , and suppose that is sufficiently large depending on . Then every subset of of...

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Rectification and the Lefschetz principle

The rectification principle in arithmetic combinatorics asserts, roughly speaking, that very small subsets (or, alternatively, small structured subsets) of an additive group or a field of large...

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An informal version of the Furstenberg correspondence principle

One of the basic objects of study in combinatorics are finite strings or infinite strings of symbols from some given alphabet , which could be either finite or infinite (but which we shall usually...

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A combinatorial subset sum problem associated with bounded prime gaps

The purpose of this post is to isolate a combinatorial optimisation problem regarding subset sums; any improvement upon the current known bounds for this problem would lead to numerical improvements...

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Expansion in finite simple groups of Lie type

Emmanuel Breuillard, Ben Green, Bob Guralnick, and I have just uploaded to the arXiv our joint paper “Expansion in finite simple groups of Lie type“. This long-delayed paper (announced way back in...

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The Poisson-Dirichlet process, and large prime factors of a random number

Define a partition of to be a finite or infinite multiset of real numbers in the interval (that is, an unordered set of real numbers in , possibly with multiplicity) whose total sum is : . For...

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Algebraic combinatorial geometry: the polynomial method in arithmetic...

I’ve just uploaded to the arXiv my article “Algebraic combinatorial geometry: the polynomial method in arithmetic combinatorics, incidence combinatorics, and number theory“, submitted to the new...

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A spectral theory proof of the algebraic regularity lemma

Let be a field. A definable set over is a set of the form where is a natural number, and is a predicate involving the ring operations of , the equality symbol , an arbitrary number of constants and...

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Ultraproducts as a Bridge Between Discrete and Continuous Analysis

(This is an extended blog post version of my talk “Ultraproducts as a Bridge Between Discrete and Continuous Analysis” that I gave at the Simons institute for the theory of computing at the workshop...

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Metric entropy analogues of sum set theory

A core foundation of the subject now known as arithmetic combinatorics (and particularly the subfield of additive combinatorics) are the elementary sum set estimates (sometimes known as “Ruzsa...

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A proof of Roth’s theorem

Roth’s theorem on arithmetic progressions asserts that every subset of the integers of positive upper density contains infinitely many arithmetic progressions of length three. There are many versions...

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Stickiness, graininess, planiness, and a sum-product approach to the Kakeya...

This is a blog version of a talk I recently gave at the IPAM workshop on “The Kakeya Problem, Restriction Problem, and Sum-product Theory”. Note: the discussion here will be highly non-rigorous in...

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Khot, Osher, Griffiths

In addition to the Fields medallists mentioned in the previous post, the IMU also awarded the Nevanlinna prize to Subhash Khot, the Gauss prize to Stan Osher (my colleague here at UCLA!), and the Chern...

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Additive limits

In graph theory, the recently developed theory of graph limits has proven to be a useful tool for analysing large dense graphs, being a convenient reformulation of the Szemerédi regularity lemma....

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Random matrices have simple spectrum

Van Vu and I have just uploaded to the arXiv our paper “Random matrices have simple eigenvalues“. Recall that an Hermitian matrix is said to have simple eigenvalues if all of its eigenvalues are...

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The Erdos-Ulam problem, varieties of general type, and the Bombieri-Lang...

In 1946, Ulam, in response to a theorem of Anning and Erdös, posed the following problem: Problem 1 (Erdös-Ulam problem) Let be a set such that the distance between any two points in is rational. Is...

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Cancellation for the multilinear Hilbert transform

I’ve just uploaded to the arXiv my paper “Cancellation for the multilinear Hilbert transform“, submitted to Collectanea Mathematica. This paper uses methods from additive combinatorics (and more...

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Nested approximate subgroups

Suppose that are two subgroups of some ambient group , with the index of in being finite. Then is the union of left cosets of , thus for some set of cardinality . The elements of are not entirely...

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