arXiv subjects

Ben Green

Publications and source records attributed to Ben Green.

At least 55 records · Page 3Linked to original sources

arXiv2017

New bounds for Szemerédi's theorem, III: A polylogarithmic bound for $r_4(N)$

Define $r_4(N)$ to be the largest cardinality of a set $A \subset \{1,\dots,N\}$ which does not contain four elements in arithmetic progression. In 1998 Gowers proved that \[ r_4(N) \ll N(\log \log N)^{-c}\] for some absolute constant $c>0$. In 2005, the authors improved this to \[ r_4(N) \ll N e^{-c\sqrt{\log\log N}}.\] In this paper we further improve this to \[ r_4(N) \ll N(\log N)^{-c},\] which appears to be the limit of our methods.

math.CO↗

arXiv2017

Sarkozy's theorem in function fields

Sárközy proved that dense sets of integers contain two elements differing by a $k$th power. The bounds in quantitative versions of this theorem are rather weak compared to what is expected. We prove a version of Sárközy's theorem for polynomials over $\mathbb{F}_q$ with polynomial dependencies in the parameters. More precisely, let $P_{q,n}$ be the space of polynomials over $\mathbb{F}_q$ of degree $< n$ in an indeterminate $T$. Let $k \geq 2$ be an integer and let $q$ be a prime power. Set $c(k,q) := (2 k^2 D_q(k)^2\log q)^{-1}$, where $D_q(k)$ is the sum of the digits of $k$ in base $q$. If $A \subset P_{q,n}$ is a set with $|A| > 2q^{(1 - c(k,q))n}$, then $A$ contains distinct polynomials $p(T), p'(T)$ such that $p(T) - p'(T) = b(T)^k$ for some $b \in \mathbb{F}_q[T]$.

math.NT↗

arXiv2017

Non-zero coefficients of half-integral weight modular forms mod $\ell$

We obtain new lower bounds for the number of Fourier coefficients of a weakly holomorphic modular form of half-integral weight not divisible by some prime $\ell$. Among the applications of this we show that there are $\gg \sqrt{X}/\log \log X$ integers $n \leq X$ for which the partition function $p(n)$ is not divisible by $\ell$, and that there are $\gg \sqrt{X}/\log \log X$ values of $n \leq X$ for which $c(n)$, the $n$th Fourier coefficient of the $j$-invariant, is odd.

math.NT↗

arXiv2017

Optical properties of Xe color centers in diamond

Optical properties of color centers in diamond have been the subject of intense research due to their promising applications in quantum photonics. In this work we study the optical properties of Xe related color centers implanted into nitrogen rich (type IIA) and an ultrapure, electronic grade diamond. The Xe defect has two zero phonon lines at ~ 794 and 811 nm, which can be effectively excited using both green and red excitation, however, its emission in the nitrogen rich diamond is brighter. Near resonant excitation is performed at cryogenic temperatures and luminescence is probed under strong magnetic field. Our results are important towards the understanding of the Xe related defect and other near infrared color centers in diamond.

cond-mat.other↗

arXiv2017

Monochromatic sums and products

Suppose that $\mathbb{F}_p$ is coloured with $r$ colours. Then there is some colour class containing at least $c_r p^2$ quadruples of the form $(x, y , x + y, xy)$.

math.NT↗

arXiv2016

Monochromatic solutions to $x + y = z^2$

Suppose that $\mathbb{N}$ is $2$-coloured. Then there are infinitely many monochromatic solutions to $x + y = z^2$. On the other hand, there is a $3$-colouring of $\mathbb{N}$ with only finitely many monochromatic solutions to this equation.

math.NT↗

arXiv2016

Invariable generation of the symmetric group

We say that permutations $π_1,\dots, π_r \in \mathcal{S}_n$ invariably generate $\mathcal{S}_n$ if, no matter how one chooses conjugates $π'_1,\dots,π'_r$ of these permutations, $π'_1,\dots,π'_r$ generate $\mathcal{S}_n$. We show that if $π_1,π_2,π_3$ are chosen randomly from $\mathcal{S}_n$ then, with probability tending to 1 as $n \rightarrow \infty$, they do not invariably generate $\mathcal{S}_n$. By contrast it was shown recently by Pemantle, Peres and Rivin that four random elements do invariably generate $\mathcal{S}_n$ with positive probability. We include a proof of this statement which, while sharing many features with their argument, is short and completely combinatorial.

math.GR↗

arXiv2016

Fourier uniformity on subspaces

Let $\mathbb{F}$ be a fixed finite field, and let $A \subset \mathbb{F}^n$. It is a well-known fact that there is a subspace $V \leq \mathbb{F}^n$, $\mbox{codim} V \ll_δ 1$, and an $x$, such that $A$ is $δ$-uniform when restricted to $x + V$ (that is, all non-trivial Fourier coefficients of $A$ restricted to $x + V$ have magnitude at most $δ$). We show that if $\mathbb{F} = \mathbb{F}_2$ then it is possible to take $x = 0$; that is, $A$ is $δ$-uniform on a subspace $V \leq \mathbb{F}^n$. We give an example to show that this is not necessarily possible when $\mathbb{F} = \mathbb{F}_3$. ADDED July 2016: shortly after this paper appeared on the arxiv, F. Manners showed us a rather short argument he had found in 2013, giving a better bound for our main theorem. We do not, therefore, intend to publish this note. The example over $\mathbb{F}_3$ may still be of interest to some readers and so we will not withdraw the paper from the arxiv.

math.NT↗

arXiv2016

On the chromatic number of random Cayley graphs

Let G be an abelian group of cardinality N, where (N,6) = 1, and let A be a random subset of G. Form a graph Gamma_A on vertex set G by joining x to y if and only if x + y is in A. Then, almost surely as N tends to infinity, the chromatic number chi(Gamma_A) is at most (1 + o(1))N/2 log_2 N. This is asymptotically sharp when G = Z/NZ, N prime. Presented at the conference in honour of Bela Bollobas on his 70th birthday, Cambridge August 2013.

math.CO↗

arXiv2016

Long gaps between primes

Let $p_n$ denotes the $n$-th prime. We prove that $$\max_{p_{n+1} \leq X} (p_{n+1}-p_n) \gg \frac{\log X \log \log X\log\log\log\log X}{\log \log \log X}$$ for sufficiently large $X$, improving upon recent bounds of the first three and fifth authors and of the fourth author. Our main new ingredient is a generalization of a hypergraph covering theorem of Pippenger and Spencer, proven using the Rödl nibble method.

math.NT↗

arXiv2015

Permutations fixing a k-set

Let $i(n,k)$ be the proportion of permutations $π\in\mathcal{S}_n$ having an invariant set of size $k$. In this note we adapt arguments of the second author to prove that $i(n,k) \asymp k^{-δ} (1+\log k)^{-3/2}$ uniformly for $1\leq k\leq n/2$, where $δ= 1 - \frac{1 + \log \log 2}{\log 2}$. As an application we show that the proportion of $π\in\mathcal{S}_n$ contained in a transitive subgroup not containing $\mathcal{A}_n$ is at least $n^{-δ+o(1)}$ if $n$ is even.

math.CO↗

arXiv2015

Large gaps between consecutive prime numbers

Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdos, we show that $$G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2},$$ where $f(X)$ is a function tending to infinity with $X$. Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes.

math.NT↗

arXiv2015

The quantitative behaviour of polynomial orbits on nilmanifolds

A theorem of Leibman asserts that a polynomial orbit $(g(1),g(2),g(3),\ldots)$ on a nilmanifold $G/Γ$ is always equidistributed in a union of closed sub-nilmanifolds of $G/Γ$. In this paper we give a quantitative version of Leibman's result, describing the uniform distribution properties of a finite polynomial orbit $(g(1),\ldots,g(N))$ in a nilmanifold. More specifically we show that there is a factorization $g = εg'γ$, where $ε(n)$ is "smooth", $γ(n)$ is periodic and "rational", and $(g'(a),g'(a+d),\ldots,g'(a + d(l-1)))$ is uniformly distributed (up to a specified error $δ$) inside some subnilmanifold $G'/Γ'$ of $G/Γ$, for all sufficiently dense arithmetic progressions $a,a+d,\ldots,a+d(l-1)$ inside $\{1,..,N\}$. Our bounds are uniform in $N$ and are polynomial in the error tolerance delta. In a subsequent paper we shall use this theorem to establish the Mobius and Nilsequences conjecture from our earlier paper "Linear equations in primes".

math.NT↗

arXiv2015

On the quantitative distribution of polynomial nilsequences - erratum

This is an erratum to 'On the quantitative distribution of polynomial nilsequences' [GT]. The proof of Theorem 8.6 of that paper, which claims a distribution result for multiparameter polynomial sequences on nilmanifolds, was incorrect. We provide two fixes for this issue here. First, we deduce the "equal sides" case $N_1 = \dots = N_t = N$ of [GT, Theorem 8.6] from the 1-parameter results in [GT]. This is the same basic mode of argument we attempted in the original paper, though the details are different. The equal sides case is the only one required in applications such as the proof of the inverse conjectures for the Gowers norms due to the authors and Ziegler. Second, we sketch a proof that [GT, Theorem 8.6] does in fact hold in its originally stated form, that is to say without the equal sides condition. To obtain this statement the entire argument of [GT] must be run in the context of multiparameter polynomial sequences $g : \mathbb{Z}^t \rightarrow G$ rather than 1-parameter sequences $g : \mathbb{Z} \rightarrow G$ as is currently done.

math.NT↗

arXiv2014

Approximate algebraic structure

We discuss a selection of recent developments in arithmetic combinatorics having to do with ``approximate algebraic structure'' together with some of their applications.

math.NT↗

arXiv2014

Bounded gaps between primes

These are notes on Zhang's work and subsequent developments produced in preparation for 5 hours of talks for a general mathematical audience given in Cambridge, Edinburgh and Auckland over the last year. Being for colloquium-style talks, these notes are at a much lower level than other accounts in the literature. In places they are deliberately quite nonrigorous. Whether anyone will find them helpful is unclear but some care was taken on their preparation and there is, I hope, little harm in making them available here.

math.NT↗

arXiv2014

Expansion in finite simple groups of Lie type

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, establishing strongly dense subgroups in simple algebraic groups.

math.GR↗

arXiv2014

Counting sets with small sumset and applications

We study the number of $k$-element sets $A \subset \{1,\ldots,N\}$ with $|A + A| \leq K|A|$ for some (fixed) $K > 0$. Improving results of the first author and of Alon, Balogh, Samotij and the second author, we determine this number up to a factor of $2^{o(k)} N^{o(1)}$ for most $N$ and $k$. As a consequence of this and a further new result concerning the number of sets $A \subset \mathbf{Z}/N\mathbf{Z}$ with $|A +A| \leq c |A|^2$, we deduce that the random Cayley graph on $\mathbf{Z}/N\mathbf{Z}$ with edge density~$\frac{1}{2}$ has no clique or independent set of size greater than $\big( 2 + o(1) \big) \log_2 N$, asymptotically the same as for the Erdős-Rényi random graph. This improves a result of the first author from 2003 in which a bound of $160 \log_2 N$ was obtained. As a second application, we show that if the elements of $A \subset \mathbf{N}$ are chosen at random, each with probability $1/2$, then the probability that $A+A$ misses exactly $k$ elements of $\mathbf{N}$ is equal to $\big( 2 + o(1) \big)^{-k/2}$ as $k \to \infty$.

math.CO↗