arXiv subjects

W. T. Gowers

Publications and source records attributed to W. T. Gowers.

At least 37 records · Page 2Linked to original sources

arXiv2017

A bilinear version of Bogolyubov's theorem

A theorem of Bogolyubov states that for every dense set $A$ in $\mathbb{Z}_N$ we may find a large Bohr set inside $A+A-A-A$. In this note, motivated by the work on a quantitative inverse theorem for the Gowers $U^4$ norm, we prove a bilinear variant of this result in vector spaces over finite fields. Namely, if we start with a dense set $A \subset \mathbb{F}^n_p \times \mathbb{F}^n_p$ and then take rows (respectively columns) of $A$ and change each row (respectively column) to the set difference of it with itself, repeating this procedure several times, we obtain a bilinear analogue of a Bohr set inside the resulting set, namely the zero set of a biaffine map from $\mathbb{F}^n_p \times \mathbb{F}^n_p$ to a $\mathbb{F}_p$-vector space of bounded dimension. An almost identical result was proved independently by Bienvenu and Lê.

math.CO↗

arXiv2016

Freiman homomorphisms on sparse random sets

A result of Fiz Pontiveros shows that if $A$ is a random subset of $\mathbb{Z}_N$ where each element is chosen independently with probability $N^{-1/2+o(1)}$, then with high probability every Freiman homomorphism defined on $A$ can be extended to a Freiman homomorphism on the whole of $\mathbb{Z}_N$. In this paper we improve the bound to $CN^{-2/3}(\log N)^{1/3}$, which is best possible up to the constant factor.

math.CO↗

arXiv2016

The length of an s-increasing sequence of r-tuples

We prove a number of results related to a problem of Po-Shen Loh, which is equivalent to a problem in Ramsey theory. Let $a=(a_1,a_2,a_3)$ and $b=(b_1,b_2,b_3)$ be two triples of integers. Define $a$ to be 2-less than $b$ if $a_i<b_i$ for at least two values of $i$, and define a sequence $a^1,\dots,a^m$ of triples to be 2-increasing if $a^r$ is 2-less than $a^s$ whenever $r<s$. Loh asks how long a 2-increasing sequence can be if all the triples take values in $\{1,2,\dots,n\}$, and gives a $\log_*$ improvement over the trivial upper bound of $n^2$ by using the triangle removal lemma. In the other direction, a simple construction gives a lower bound of $n^{3/2}$. We look at this problem and a collection of generalizations, improving some of the known bounds, pointing out connections to other well known problems in extremal combinatorics, and asking a number of further questions.

math.CO↗

arXiv2016

Inverse and stability theorems for approximate representations of finite groups

The $U^2$ norm gives a useful measure of quasirandomness for real- or complex-valued functions defined on finite (or, more generally, locally compact) groups. A simple Fourier-analytic argument yields an inverse theorem, which shows that a bounded function with a large $U^2$ norm defined on a finite Abelian group must correlate significantly with a character. In this paper we generalize this statement to functions that are defined on arbitrary finite groups and that take values in M$_n(\mathbb C)$. The conclusion now is that the function correlates with a representation -- though with the twist that the dimension of the representation is shown to be within a constant of $n$ rather than being exactly equal to $n$. There are easy examples that show that this weakening of the obvious conclusion is necessary. The proof is much less straightforward than it is in the case of scalar functions on Abelian groups. As an easy corollary, we prove a stability theorem for near representations. It states that if $G$ is a finite group and $f:G\to$M$_n(\mathbb C)$ is a function that is close to a representation in the sense that $f(xy)-f(x)f(y)$ has a small Hilbert-Schmidt norm (also known as the Frobenius norm) for every $x,y\in G$, then there must be a representation $ρ$ such that $f(x)-ρ(x)$ has small Hilbert-Schmidt norm for every $x$. Again, the dimension of $ρ$ need not be exactly $n$, but it must be close to $n$. We also obtain stability theorems for other Schatten $p$-norms. A stability theorem of this kind was obtained for the operator norm by Grove, Karcher and Ruh in 1974 and in a more general form by Kazhdan in 1982. (For the operator norm, the dimension of the approximating representation is exactly $n$.)

math.GR↗

arXiv2016

Generalizations of Fourier analysis, and how to apply them

This is a survey of the use of Fourier analysis in additive combinatorics, with a particular focus on situations where it cannot be straightforwardly applied, but needs to be generalized first. Sometimes very satisfactory generalizations exist, while sometimes we have to make do with theories that have some of the desirable properties of Fourier analysis but not all of them. In the latter case, there are intriguing hints that there may be more satisfactory theories yet to be discovered. This article grew out of the Colloquium Lectures at the Joint Meeting of the AMS and the MAA, given in Seattle in January 2016.

math.CO↗

arXiv2015

Erdős and Arithmetic Progressions

This is a short survey article written for the Erdős centennial conference in Budapest in 2013. The main two topics covered are Szemerédi's theorem and its ramifications, and the Erdős discrepancy problem. There is an emphasis on what we do not yet know, so much of the article is somewhat speculative.

math.CO↗

arXiv2015

Combinatorial theorems in sparse random sets

We develop a new technique that allows us to show in a unified way that many well-known combinatorial theorems, including Turán's theorem, Szemerédi's theorem and Ramsey's theorem, hold almost surely inside sparse random sets. For instance, we extend Turán's theorem to the random setting by showing that for every $ε> 0$ and every positive integer $t \geq 3$ there exists a constant $C$ such that, if $G$ is a random graph on $n$ vertices where each edge is chosen independently with probability at least $C n^{-2/(t+1)}$, then, with probability tending to $1$ as $n$ tends to infinity, every subgraph of $G$ with at least $(1 - \frac{1}{t-1} + ε) e(G)$ edges contains a copy of $K_t$. This is sharp up to the constant $C$. We also show how to prove sparse analogues of structural results, giving two main applications, a stability version of the random Turán theorem stated above and a sparse hypergraph removal lemma. Many similar results have recently been obtained independently in a different way by Schacht and by Friedgut, Rödl and Schacht.

math.CO↗

arXiv2013

A fully automatic problem solver with human-style output

This paper describes a program that solves elementary mathematical problems, mostly in metric space theory, and presents solutions that are hard to distinguish from solutions that might be written by human mathematicians. The program is part of a more general project, which we also discuss.

cs.AI↗

arXiv2013

On the KŁR conjecture in random graphs

The KŁR conjecture of Kohayakawa, Łuczak, and Rödl is a statement that allows one to prove that asymptotically almost surely all subgraphs of the random graph G_{n,p}, for sufficiently large p : = p(n), satisfy an embedding lemma which complements the sparse regularity lemma of Kohayakawa and Rödl. We prove a variant of this conjecture which is sufficient for most known applications to random graphs. In particular, our result implies a number of recent probabilistic versions, due to Conlon, Gowers, and Schacht, of classical extremal combinatorial theorems. We also discuss several further applications.

math.CO↗

arXiv2010

Linear forms and higher-degree uniformity for functions on $\mathbb{F}_p^n$

In [GW09a] we conjectured that uniformity of degree $k-1$ is sufficient to control an average over a family of linear forms if and only if the $k$th powers of these linear forms are linearly independent. In this paper we prove this conjecture in $\mathbb{F}_p^n$, provided only that $p$ is sufficiently large. This result represents one of the first applications of the recent inverse theorem for the $U^k$ norm over $\mathbb{F}_p^n$ by Bergelson, Tao and Ziegler [BTZ09,TZ08]. We combine this result with some abstract arguments in order to prove that a bounded function can be expressed as a sum of polynomial phases and a part that is small in the appropriate uniformity norm. The precise form of this decomposition theorem is critical to our proof, and the theorem itself may be of independent interest.

math.NT↗

arXiv2010

Linear forms and quadratic uniformity for functions on $\mathbb{Z}_N$

A very useful fact in additive combinatorics is that analytic expressions that can be used to count the number of structures of various kinds in subsets of Abelian groups are robust under quasirandom perturbations, and moreover that quasirandomness can often be measured by means of certain easily described norms, known as uniformity norms. However, determining which uniformity norms work for which structures turns out to be a surprisingly hard question. In [GW09a] and [GW09b, GW09c] we gave a complete answer to this question for groups of the form $G=\mathbb{F}_p^n$, provided $p$ is not too small. In $\mathbb{Z}_N$, substantial extra difficulties arise, of which the most important is that an "inverse theorem" even for the uniformity norm $\|.\|_{U^3}$ requires a more sophisticated (local) formulation. When $N$ is prime, $\mathbb{Z}_N$ is not rich in subgroups, so one must use regular Bohr neighbourhoods instead. In this paper, we prove the first non-trivial case of the main conjecture from [GW09a].

math.NT↗

arXiv2010

Linear forms and quadratic uniformity for functions on $\mathbb{F}_p^n$

We give improved bounds for our theorem in [GW09], which shows that a system of linear forms on $\mathbb{F}_p^n$ with squares that are linearly independent has the expected number of solutions in any linearly uniform subset of $\mathbb{F}_p^n$. While in [GW09] the dependence 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 relation between the two parameters. Instead of the structure theorem for bounded functions due to Green and Tao [GrT08], we use the Hahn-Banach theorem to decompose the function into a quadratically structured plus a quadratically uniform part. This new decomposition makes more efficient use of the $U^3$ inverse theorem [GrT08].

math.NT↗

arXiv2008

Decompositions, approximate structure, transference, and the Hahn-Banach theorem

This paper is partly a survey of certain kinds of results and proofs in additive combinatorics, and partly a discussion of how useful the finite-dimensional Hahn-Banach theorem can be. The most interesting single result is probably a simpler proof of a key step in the proof of the Green-Tao theorem, but several other applications of the method are given. A similarly simplified proof of the Green-Tao transference principle was obtained independently (and expressed in a rather different language) by Reingold, Trevisan, Tulsiani and Vadhan.

math.CO↗

arXiv2007

The true complexity of a system of linear equations

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, 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.

math.NT↗

arXiv2007

Quasirandom groups

Babai and Sós have asked whether there exists a constant c>0 such that every finite group G has a product-free subset of size at least c|G|: that is, a subset X that does not contain three elements x, y and z with xy=z. In this paper we show that the answer is no. Moreover, we give a simple sufficient condition for a group not to have any large product-free subset.

math.CO↗

arXiv2007

Hypergraph regularity and the multidimensional Szemerédi theorem

We prove analogues for hypergraphs of Szemerédi's regularity lemma and the associated counting lemma for graphs. As an application, we give the first combinatorial proof of the multidimensional Szemerédi theorem of Furstenberg and Katznelson, and the first proof that provides an explicit bound. Similar results with the same consequences have been obtained independently by Nagle, Rödl, Schacht and Skokan.

math.CO↗

arXiv2005

An infinite Ramsey theorem and some Banach-space dichotomies

A problem of Banach asks whether every infinite-dimensional Banach space which is isomorphic to all its infinite-dimensional subspaces must be isomorphic to a separable Hilbert space. In this paper we prove a result of a Ramsey-theoretic nature which implies an interesting dichotomy for subspaces of Banach spaces. Combined with a result of Komorowski and Tomczak-Jaegermann, this gives a positive answer to Banach's problem. We then generalize the Ramsey-theoretic result and deduce a further dichotomy for Banach spaces with an unconditional basis.

math.FA↗

arXiv1994

Banach spaces with small spaces of operators

For a certain class of algebras $\cal A$ we give a method for constructing Banach spaces $X$ such that every operator on $X$ is close to an operator in $\cal A$. This is used to produce spaces with a small amount of structure. We present several applications. Amongst them are constructions of a new prime Banach space, a space isomorphic to its subspaces of codimension two but not to its hyperplanes and a space isomorphic to its cube but not to its square.

math.FA↗