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.

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.