5,627 research outputs found
Freiman's theorem in finite fields via extremal set theory
Using various results from extremal set theory (interpreted in the language
of additive combinatorics), we prove an asyptotically sharp version of
Freiman's theorem in F_2^n: if A in F_2^n is a set for which |A + A| <= K|A|
then A is contained in a subspace of size 2^{2K + O(\sqrt{K}\log K)}|A|; except
for the O(\sqrt{K} \log K) error, this is best possible. If in addition we
assume that A is a downset, then we can also cover A by O(K^{46}) translates of
a coordinate subspace of size at most |A|, thereby verifying the so-called
polynomial Freiman-Ruzsa conjecture in this case. A common theme in the
arguments is the use of compression techniques. These have long been familiar
in extremal set theory, but have been used only rarely in the additive
combinatorics literature.Comment: 18 page
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.Comment: 23 pages, survey article submitted to Proceedings of the Erdos
Centenary conferenc
A note on the Freiman and Balog-Szemeredi-Gowers theorems in finite fields
We obtain quantitative versions of the Balog-Szemeredi-Gowers and Freiman
theorems in the model case of a finite field geometry F_2^n, improving the
previously known bounds in such theorems. For instance, if A is a subset of
F_2^n such that |A+A| <= K|A| (thus A has small additive doubling), we show
that there exists an affine subspace V of F_2^n of cardinality |V| >>
K^{-O(\sqrt{K})} |A| such that |A \cap V| >> |V|/2K. Under the assumption that
A contains at least |A|^3/K quadruples with a_1 + a_2 + a_3 + a_4 = 0 we obtain
a similar result, albeit with the slightly weaker condition |V| >>
K^{-O(K)}|A|.Comment: 12 pages, to appear in J. Aust. Math. Society. Some very minor
revisions from previous versio
Self similar sets, entropy and additive combinatorics
This article is an exposition of recent results on self-similar sets,
asserting that if the dimension is smaller than the trivial upper bound then
there are almost overlaps between cylinders. We give a heuristic derivation of
the theorem using elementary arguments about covering numbers. We also give a
short introduction to additive combinatorics, focusing on inverse theorems,
which play a pivotal role in the proof. Our elementary approach avoids many of
the technicalities in the original proof but also falls short of a complete
proof. In the last section we discuss how the heuristic argument is turned into
a rigorous one.Comment: 21 pages, 2 figures; submitted to Proceedings of AFRT 2012. v5: more
typos correcte
Combinatorial Applications of the Subspace Theorem
The Subspace Theorem is a powerful tool in number theory. It has appeared in
various forms and been adapted and improved over time. It's applications
include diophantine approximation, results about integral points on algebraic
curves and the construction of transcendental numbers. But its usefulness
extends beyond the realms of number theory. Other applications of the Subspace
Theorem include linear recurrence sequences and finite automata. In fact, these
structures are closely related to each other and the construction of
transcendental numbers.
The Subspace Theorem also has a number of remarkable combinatorial
applications. The purpose of this paper is to give a survey of some of these
applications including sum-product estimates and bounds on unit distances. The
presentation will be from the point of view of a discrete mathematician. We
will state a number of variants of the Subspace Theorem below but we will not
prove any of them as the proofs are beyond the scope of this work. However we
will give a proof of a simplified special case of the Subspace Theorem which is
still very useful for many problems in discrete mathematics
An ansatz for the asymptotics of hypergeometric multisums
Sequences that are defined by multisums of hypergeometric terms with compact
support occur frequently in enumeration problems of combinatorics, algebraic
geometry and perturbative quantum field theory. The standard recipe to study
the asymptotic expansion of such sequences is to find a recurrence satisfied by
them, convert it into a differential equation satisfied by their generating
series, and analyze the singulatiries in the complex plane. We propose a
shortcut by constructing directly from the structure of the hypergeometric term
a finite set, for which we conjecture (and in some cases prove) that it
contains all the singularities of the generating series. Our construction of
this finite set is given by the solution set of a balanced system of polynomial
equations of a rather special form, reminiscent of the Bethe ansatz. The finite
set can also be identified with the set of critical values of a potential
function, as well as with the evaluation of elements of an additive -theory
group by a regulator function. We give a proof of our conjecture in some
special cases, and we illustrate our results with numerous examples.Comment: 22 pages and 2 figure
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