Mathematics - Preprints

For a general description of my area of research, please visit the section Mathematics - Research Interests, or have a look at Mathematics - A Personal Roadmap.

Papers

'The true complexity of a system of linear equations' with W.T. Gowers

Abstract:	It is well-known that if a subset A of a finite Abelian group G satisfies a quasirandomness property called uniformity of degree k, then it contains roughly the expected number of arithmetic progressions of length k+2, that is, the number of progressions one would expect in a random subset of G of the same density as A. One is naturally led to ask which degree of uniformity is required of A in order to control the number of solutions to a general system of linear equations. Using so-called quadratic Fourier analysis, we show that certain linear systems that were previously thought to require quadratic uniformity are in fact governed by linear uniformity. More generally, we conjecture a necessary and sufficient condition on a linear system L which guarantees that any subset A of F_p^n which is uniform of degree k contains the expected number of solutions to L.
Status:	accepted, Proc. Lond. Math. Soc.
Last updated:	1st November 2007
Download:	arXiv:0711.0185

'A note on Elkin's improvement of Behrend's construction' with B.J. Green

Abstract:	We provide a short proof of a recent result of Elkin [E08] in which large subsets of {1, 2, ..., N} free of 3-term progressions are constructed.
Status:	submitted
Last updated:	3rd October 2008
Download:	arXiv:0810.0732

'The structure of popular difference sets'

Abstract:	We show that the set of popular differences of a large subset of Z_N does not always contain the complete difference set of another large set. For this purpose we construct a so-called niveau set, which was first introduced by Ruzsa in [Ru87] and later used in [Ru91] to show that there exists a large subset of Z_N whose sumset does not contain any long arithmetic progressions. In this paper we make substantial use of measure concentration results on the multi-dimensional torus and Esseen's Inequality.
Status:	submitted
Last updated:	1st June 2008
Download:	available soon, see the note by Tom Sanders at arXiv:0807.5106

'The minimal number of monochromatic 4-term progressions'

Abstract:	We improve the lower bound given by Cameron, Cilleruelo and Serra [CSS05] for the minimum number of monochromatic 4-term progressions contained in any 2-colouring of Z_p with p a prime. We also exhibit a colouring with significantly fewer than the random number of monochromatic 4-term progressions, which is based on an a recent example in additive combinatorics by Gowers [Gow06]. In the second half of this paper, we discuss the corresponding problem in graphs, which has received a great deal more attention to date. We give a simplified proof of the best known lower bound on the minimum number of monochromatic K_4s contained in any 2-colouring of K_n by Giraud [Gir79], and briefly discuss the analogy between the upper-bound graph constructions of Thomason [Tho89] and ours for subsets of Z_p.
Status:	preprint
Last updated:	1st December 2007
Download:	available soon

'Decompositions into polynomial phase functions' with W.T. Gowers

Abstract:	We give improved bounds for our theorem in [GW07], which stated that square-independent systems over F_p^n (of Cauchy-Schwarz complexity 2) are governed by ordinary uniformity, that is, any uniform subset of F_p^n contains the expected number of solutions to the linear system. While in [GW07] the dependency between the uniformity of the set and the resulting error in the average over the linear system was of tower-type, we now obtain a doubly exponential dependence. This is achieved using an alternative version of the structure theorem by Green and Tao [GT05], which decomposes any bounded function into a 'quadratically structured' plus a quadratically uniform part. This new decomposition makes more efficient use of the U^3 inverse theorem and is likely to find further applications in arithmetic combinatorics. The second part of the paper deals with the corresponding problem in the cyclic group Z_N with N a prime. While the main ideas of the proof for the F_p^n case carry over, substantial technical difficulties arise due to the lack of rigid algebraic substructures in Z_N. We need to use so-called Bohr sets as approximated subgroups, together with a number of tools and techniques pioneered by Bourgain [Bou99].
Status:	in preparation
Last updated:	1st October 2008
Download:	available soon

'Minimal characteristic factors for linear systems'

Abstract:	In this expository note we outline the analogies between two recent preprints by Leibman [L07] and Gowers and the author [GW07]. Both papers independently describe two manifestations of the same phenomenon, the former in the context of ergodic theory and the latter in arithmetic combinatorics. In their respective settings, they address the question after the degree of the minimal characteristic factor of a multiple ergodic average along a system of linear forms, or the minimal degree of uniformity needed to accurately count solutions to the corresponding system of linear equations. The present article is aimed at readers with a combinatorics background and limited prior exposure to ergodic theory.
Last updated:	1st September 2007
Download:	pdf

'The inverse theorem over F_2^n'

Abstract:	This is a short note for the purpose of establishing a local version of the U^3 inverse theorem in the special case of F_2^n following very closely the original argument of Samorodnitsky [S07].
Last updated:	1st October 2008
Download:	available soon

'Sets whose difference set is square-free'

Abstract:	The purpose of this article is to give an exposition of the best-known bound on the density of sets whose difference set contains no squares which was first derived by Pintz, Steiger and Szemerédi in [PSS88]. We show how this method can be brought in line with the modern view of the energy increment strategy employed in problems such as Szemerédi's Theorem on arithmetic progressions, and explore the extent to which the particularities of the method are specific to the set of squares.
Last updated:	8th July 2005
Download:	available soon

'Arithmetic structures in difference sets' under the supervision of B.J. Green

Abstract:	A well-known Theorem of Sarközy states that provided a subset of the set {1,...,N} is dense enough, it contains two elements whose difference is a perfect square. Recently an elegant proof (involving elementary Fourier analysis) was given by Green, and it is on this result that most of the essay is based. In fact, we give a new combinatorial proof of an (already established) extension of Sarközy's result to cover the case where 'perfect square' is replaced by any quadratic polynomial with an integer root. We make heavy use of the so-called 'circle method' developed by Hardy and Littlewood, which we describe in some detail. As an application we also discuss the asymptotic solution to Waring's problem. The opening chapter includes a survey of some classical results on exponential sums such as Weyl's Inequality and Hua's Lemma.
Last updated:	23rd May 2003
Download:	pdf

This page was last updated 1st October 2008.