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1. CMB 2011 (vol 56 pp. 412)
Structure in Sets with Logarithmic Doubling Suppose that $G$ is an abelian group, $A \subset G$ is finite with $|A+A| \leq K|A|$ and $\eta \in (0,1]$ is a parameter.
Our main result is that there is a set $\mathcal{L}$ such that
\begin{equation*}
|A \cap \operatorname{Span}(\mathcal{L})| \geq K^{-O_\eta(1)}|A| \quad\text{and}\quad |\mathcal{L}| = O(K^\eta\log |A|).
\end{equation*}
We include an application of this result to a generalisation of the Roth--Meshulam theorem due to Liu and Spencer.
Keywords:Fourier analysis, Freiman's theorem, capset problem Category:11B25 |
2. CMB 1999 (vol 42 pp. 25)
On the Set of Common Differences in van der Waerden's Theorem on Arithmetic Progressions Analogues of van der Waerden's theorem on arithmetic progressions
are considered where the family of all arithmetic progressions,
$\AP$, is replaced by some subfamily of $\AP$. Specifically, we
want to know for which sets $A$, of positive integers, the
following statement holds: for all positive integers $r$ and $k$,
there exists a positive integer $n= w'(k,r)$ such that for every
$r$-coloring of $[1,n]$ there exists a monochromatic $k$-term
arithmetic progression whose common difference belongs to $A$. We
will call any subset of the positive integers that has the above
property {\em large}. A set having this property for a specific
fixed $r$ will be called {\em $r$-large}. We give some necessary
conditions for a set to be large, including the fact that every
large set must contain an infinite number of multiples of each
positive integer. Also, no large set $\{a_{n}: n=1,2,\dots\}$ can
have $\liminf\limits_{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}} > 1$.
Sufficient conditions for a set to be large are also given. We
show that any set containing $n$-cubes for arbitrarily large $n$,
is a large set. Results involving the connection between the
notions of ``large'' and ``2-large'' are given. Several open
questions and a conjecture are presented.
Categories:11B25, 05D10 |