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Ben Green

Publications and source records attributed to Ben Green.

At least 73 records · Page 4Linked to original sources
arXiv2013

On sets defining few ordinary lines

Let P be a set of n points in the plane, not all on a line. We show that if n is large then there are at least n/2 ordinary lines, that is to say lines passing through exactly two points of P. This confirms, for large n, a conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for this problem, as well as all sets having fewer than n - C ordinary lines for some absolute constant C. We also solve, for large n, the "orchard-planting problem", which asks for the maximum number of lines through exactly 3 points of P. Underlying these results is a structure theorem which states that if P has at most Kn ordinary lines then all but O(K) points of P lie on a cubic curve, if n is sufficiently large depending on K.

math.CO↗
arXiv2013

Small doubling in groups

Let A be a subset of a group G = (G,.). We will survey the theory of sets A with the property that |A.A| <= K|A|, where A.A = {a_1 a_2 : a_1, a_2 in A}. The case G = (Z,+) is the famous Freiman--Ruzsa theorem.

math.NT↗
arXiv2012

On (not) computing the Mobius function using bounded depth circuits

Any function F : {0,...,N-1} -> {-1,1} such that F(x) can be computed from the binary digits of x using a bounded depth circuit is orthogonal to the Mobius function mu in the sense that E_{0 <= x <= N-1} mu(x)F(x) = o(1). The proof combines a result of Linial, Mansour and Nisan with techniques of Katai and Harman-Katai, used in their work on finding primes with specified digits.

math.NT↗
arXiv2012

New bounds for Szemeredi's theorem, Ia: Progressions of length 4 in finite field geometries revisited

Let p > 4 be a prime. We show that the largest subset of F_p^n with no 4-term arithmetic progressions has cardinality << N(log N)^{-c}, where c = 2^{-22} and N := p^n. A result of this type was claimed in a previous paper by the authors and published in Proc. London Math. Society. Unfortunately the proof had a gap, and we issue an erratum for that paper here. Our new argument is different and significantly shorter. In fact we prove a stronger result, which can be viewed as a quantatitive version of some previous results of Bergelson-Host-Kra and the authors.

math.NT↗
arXiv2011

A nilpotent Freiman dimension lemma

We prove that a K-approximate subgroup of an arbitrary torsion-free nilpotent group can be covered by a bounded number of cosets of a nilpotent subgroup of bounded rank, where the bounds are explicit and depend only on K. The result can be seen as a nilpotent analogue to Freiman's dimension lemma.

math.CO↗
arXiv2011

Contractions and expansion

Let A be a finite set of reals and let K >= 1 be a real number. Suppose that for each a in A we are given an injective map f_a : A -> R which fixes a and contracts other points towards it in the sense that |a - f_a(x)| <= |a - x|/K for all x in A, and such that f_a(x) always lies between a and x. Then the union of the f_a(A) has cardinality >= K|A|/10 - O_K(1). An immediate consequence of this is the estimate |A + K.A| >= K|A|/10 - O_K(1), which is a slightly weakened version of a result of Bukh.

math.CO↗
arXiv2011

The structure of approximate groups

Let K >= 1 be a parameter. A K-approximate group is a finite set A in a (local) group which contains the identity, is symmetric, and such that A^2 is covered by K left translates of A. The main result of this paper is a qualitative description of approximate groups as being essentially finite-by-nilpotent, answering a conjecture of H. Helfgott and E. Lindenstrauss. This may be viewed as a generalisation of the Freiman-Ruzsa theorem on sets of small doubling in the integers to arbitrary groups. We begin by establishing a correspondence principle between approximate groups and locally compact (local) groups that allows us to recover many results recently established in a fundamental paper of Hrushovski. In particular we establish that approximate groups can be approximately modeled by Lie groups. To prove our main theorem we apply some additional arguments essentially due to Gleason. These arose in the solution of Hilbert's fifth problem in the 1950s. Applications of our main theorem include a finitary refinement of Gromov's theorem, as well as a generalized Margulis lemma conjectured by Gromov and a result on the virtual nilpotence of the fundamental group of Ricci almost nonnegatively curved manifolds.

math.GR↗
arXiv2011

An inverse theorem for the Gowers U^{s+1}[N]-norm (announcement)

In this note we announce the proof of the inverse conjecture for the Gowers U^{s+1}[N]-norm for all s => 3; this is new for s => 4, the cases s = 1,2,3 having been previously established. More precisely we outline a proof (details of which will appear in a forthcoming paper) that if f : [N] -> [-1,1] is a function with || f ||_{U^{s+1}[N]} => δthen there is a bounded-complexity s-step nilsequence F(g(n)Γ) which correlates with f, where the bounds on the complexity and correlation depend only on s and δ. From previous results, this conjecture implies the Hardy-Littlewood prime tuples conjecture for any linear system of finite complexity. In particular, one obtains an asymptotic formula for the number of k-term arithmetic progressions p_1 < p_2 < ... < p_k <= N of primes, for every k => 3.

math.NT↗
arXiv2011

Strongly dense free subgroups of semisimple algebraic groups

We show that (with one possible exception) there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski dense. As a consequence, we get new generating results for finite simple groups of Lie type and a strengthening of a theorem of Borel related to the Hausdorff-Banach-Tarski paradox. In a sequel to this paper, we use this result to also establish uniform expansion properties for random Cayley graphs over finite simple groups of Lie type.

math.GR↗
arXiv2011

Approximate groups, III: the unitary case

By adapting the classical proof of Jordan's theorem on finite subgroups of linear groups, we show that every approximate subgroup of the unitary group U_n(C) is almost abelian.

math.GR↗
arXiv2010

An inverse theorem for the Gowers U^4 norm

We prove the so-called inverse conjecture for the Gowers U^{s+1}-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we establish that if f : [N] -> C is a function with |f(n)| <= 1 for all n and || f ||_{U^4} >= δthen there is a bounded complexity 3-step nilsequence F(g(n)Γ) which correlates with f. The approach seems to generalise so as to prove the inverse conjecture for s >= 4 as well, and a longer paper will follow concerning this. By combining this with several previous papers of the first two authors one obtains the generalised Hardy-Littlewood prime-tuples conjecture for any linear system of complexity at most 3. In particular, we have an asymptotic for the number of 5-term arithmetic progressions p_1 < p_2 < p_3 < p_4 < p_5 <= N of primes.

math.NT↗
arXiv2010

Yet another proof of Szemeredi's theorem

Using the density-increment strategy of Roth and Gowers, we derive Szemeredi's theorem on arithmetic progressions from the inverse conjectures GI(s) for the Gowers norms, recently established by the authors and Ziegler.

math.NT↗
arXiv2010

Approximate subgroups of linear groups

We establish various results on the structure of approximate subgroups in linear groups such as SL_n(k) that were previously announced by the authors. For example, generalising a result of Helfgott (who handled the cases n = 2 and 3), we show that any approximate subgroup of SL_n(F_q) which generates the group must be either very small or else nearly all of SL_n(F_q). The argument generalises to other absolutely almost simple connected (and non-commutative) algebraic groups G over a finite field k. In a subsequent paper, we will give applications of this result to the expansion properties of Cayley graphs.

math.GR↗
arXiv2010

Suzuki groups as expanders

We show that pairs of generators for the family Sz(q) of Suzuki groups may be selected so that the corresponding Cayley graphs are expanders. By combining this with several deep works of Kassabov, Lubotzky and Nikolov, this establishes that the family of all non-abelian finite simple groups can be made into expanders in a uniform fashion.

math.GR↗
arXiv2010

Boolean functions with small spectral norm

Suppose that f is a boolean function from F_2^n to {0,1} with spectral norm (that is the sum of the absolute values of its Fourier coefficients) at most M. We show that f may be expressed as +/- 1 combination of at most 2^(2^(O(M^4))) indicator functions of subgroups of F_2^n.

math.CA↗
arXiv2010

Linear Approximate Groups

This is an informal announcement of results to be described and proved in detail in a paper to appear. We give various results on the structure of approximate subgroups in linear groups such as $\SL_n(k)$. For example, generalising a result of Helfgott (who handled the cases $n = 2$ and 3), we show that any approximate subgroup of $\SL_n(\F_q)$ which generates the group must be either very small or else nearly all of $\SL_n(\F_q)$. The argument is valid for all Chevalley groups $G(\F_q)$.

math.GR↗
arXiv2009

Approximate groups and their applications: work of Bourgain, Gamburd, Helfgott and Sarnak

This is a survey of several exciting recent results in which techniques originating in the area known as additive combinatorics have been applied to give results in other areas, such as group theory, number theory and theoretical computer science. We begin with a discussion of the notion of an approximate group and also that of an approximate field, describing key results of Freiman-Ruzsa, Bourgain-Katz-Tao, Helfgott and others in which the structure of such objects is elucidated. We then move on to the applications. In particular we will look at the work of Bourgain and Gamburd on expansion properties of Cayley graphs on SL_2(F_p) and at its application in the work of Bourgain, Gamburd and Sarnak on nonlinear sieving problems.

math.NT↗