167 research outputs found
The Freiman-Ruzsa Theorem in Finite Fields
Abstract Let G be a finite abelian group of torsion r and let A be a subset of G. The Freiman-Ruzsa theorem asserts that if |A + A| ≤ K|A| then A is contained in a coset of a subgroup of G of size at most K 2 r K 4 |A|. It was conjectured by Ruzsa that the subgroup size can be reduced to r CK |A| for some absolute constant C ≥ 2. This conjecture was verified for r = 2 in a sequence of recent works, which have, in fact, yielded a tight bound. In this work, we establish the same conjecture for any prime torsion
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
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.Comment: 25 pages. Submitted to Proceedings of the ICM 2014. This version may
be longer than the published one, as my submission was 4 pages too long with
the official style fil
The Freiman--Ruzsa Theorem over Finite Fields
Let G be a finite abelian group of torsion r and let A be a subset of G. The
Freiman--Ruzsa theorem asserts that if |A+A| < K|A| then A is contained in a
coset of a subgroup of G of size at most r^{K^4}K^2|A|. It was conjectured by
Ruzsa that the subgroup size can be reduced to r^{CK}|A| for some absolute
constant C >= 2. This conjecture was verified for r = 2 in a sequence of recent
works, which have, in fact, yielded a tight bound. In this work, we establish
the same conjecture for any prime torsion
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
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