73 research outputs found
A probabilistic technique for finding almost-periods of convolutions
We introduce a new probabilistic technique for finding 'almost-periods' of
convolutions of subsets of groups. This gives results similar to the
Bogolyubov-type estimates established by Fourier analysis on abelian groups but
without the need for a nice Fourier transform to exist. We also present
applications, some of which are new even in the abelian setting. These include
a probabilistic proof of Roth's theorem on three-term arithmetic progressions
and a proof of a variant of the Bourgain-Green theorem on the existence of long
arithmetic progressions in sumsets A+B that works with sparser subsets of {1,
..., N} than previously possible. In the non-abelian setting we exhibit
analogues of the Bogolyubov-Freiman-Halberstam-Ruzsa-type results of additive
combinatorics, showing that product sets A B C and A^2 A^{-2} are rather
structured, in the sense that they contain very large iterated product sets.
This is particularly so when the sets in question satisfy small-doubling
conditions or high multiplicative energy conditions. We also present results on
structures in product sets A B. Our results are 'local' in nature, meaning that
it is not necessary for the sets under consideration to be dense in the ambient
group. In particular, our results apply to finite subsets of infinite groups
provided they 'interact nicely' with some other set.Comment: 29 pages, to appear in GAF
Approximate groups, II: the solvable linear case
We describe the structure of "K-approximate subgroups'' of solvable subgroups
of GL_n(C), showing that they have a large nilpotent piece. By combining this
with the main result of our recent paper on approximate subgroups of
torsion-free nilpotent groups, we show that such approximate subgroups are
efficiently controlled by nilpotent progressions.Comment: 10 page
On a non-abelian Balog-Szemeredi-type lemma
We show that if G is a group and A is a finite subset of G with |A^2| < K|A|,
then for all k there is a symmetric neighbourhood of the identity S with S^k a
subset of A^2A^{-2} and |S| > exp(-K^{O(k)})|A|.Comment: 5 pp. Corrected typos. Minor revision
Freiman's theorem for solvable groups
Freiman's theorem asserts, roughly speaking, if that a finite set in a
torsion-free abelian group has small doubling, then it can be efficiently
contained in (or controlled by) a generalised arithmetic progression. This was
generalised by Green and Ruzsa to arbitrary abelian groups, where the
controlling object is now a coset progression. We extend these results further
to solvable groups of bounded derived length, in which the coset progressions
are replaced by the more complicated notion of a "coset nilprogression". As one
consequence of this result, any subset of such a solvable group of small
doubling is is controlled by a set whose iterated products grow polynomially,
and which are contained inside a virtually nilpotent group. As another
application we establish a strengthening of the Milnor-Wolf theorem that all
solvable groups of polynomial growth are virtually nilpotent, in which only one
large ball needs to be of polynomial size. This result complements recent work
of Breulliard-Green, Fisher-Katz-Peng, and Sanders.Comment: 41 pages, no figures, to appear, Contrib. Disc. Math. More discussion
and examples added, as per referee suggestions; also references to subsequent
wor
On the Bogolyubov-Ruzsa lemma
Our main result is that if A is a finite subset of an abelian group with
|A+A| < K|A|, then 2A-2A contains an O(log^{O(1)} K)-dimensional coset
progression M of size at least exp(-O(log^{O(1)} K))|A|.Comment: 28 pp. Corrected typos. Added appendix on model settin
Which groups are amenable to proving exponent two for matrix multiplication?
The Cohn-Umans group-theoretic approach to matrix multiplication suggests
embedding matrix multiplication into group algebra multiplication, and bounding
in terms of the representation theory of the host group. This
framework is general enough to capture the best known upper bounds on
and is conjectured to be powerful enough to prove , although
finding a suitable group and constructing such an embedding has remained
elusive. Recently it was shown, by a generalization of the proof of the Cap Set
Conjecture, that abelian groups of bounded exponent cannot prove
in this framework, which ruled out a family of potential constructions in the
literature.
In this paper we study nonabelian groups as potential hosts for an embedding.
We prove two main results:
(1) We show that a large class of nonabelian groups---nilpotent groups of
bounded exponent satisfying a mild additional condition---cannot prove in this framework. We do this by showing that the shrinkage rate of powers
of the augmentation ideal is similar to the shrinkage rate of the number of
functions over that are degree polynomials;
our proof technique can be seen as a generalization of the polynomial method
used to resolve the Cap Set Conjecture.
(2) We show that symmetric groups cannot prove nontrivial bounds on
when the embedding is via three Young subgroups---subgroups of the
form ---which is a
natural strategy that includes all known constructions in .
By developing techniques for negative results in this paper, we hope to
catalyze a fruitful interplay between the search for constructions proving
bounds on and methods for ruling them out.Comment: 23 pages, 1 figur
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.Comment: 25 pages. Survey article to accompany my forthcoming talk at the
Current Events Bulletin of the AMS, 2010. A reference added and a few small
changes mad
Optimal Inverse Littlewood-Offord theorems
Let eta_i be iid Bernoulli random variables, taking values -1,1 with
probability 1/2. Given a multiset V of n integers v_1,..., v_n, we define the
concentration probability as rho(V) := sup_{x} Pr(v_1 eta_1+...+ v_n eta_n=x).
A classical result of Littlewood-Offord and Erdos from the 1940s asserts that
if the v_i are non-zero, then rho(V) is O(n^{-1/2}). Since then, many
researchers obtained improved bounds by assuming various extra restrictions on
V. About 5 years ago, motivated by problems concerning random matrices, Tao and
Vu introduced the Inverse Littlewood-Offord problem. In the inverse problem,
one would like to give a characterization of the set V, given that rho(V) is
relatively large. In this paper, we introduce a new method to attack the
inverse problem. As an application, we strengthen a previous result of Tao and
Vu, obtaining an optimal characterization for V. This immediately implies
several classical theorems, such as those of Sarkozy-Szemeredi and Halasz. The
method also applies in the continuous setting and leads to a simple proof for
the beta-net theorem of Tao and Vu, which plays a key role in their recent
studies of random matrices. All results extend to the general case when V is a
subset of an abelian torsion-free group and eta_i are independent variables
satisfying some weak conditions
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