Suppose $\Gamma$ is a finitely generated countable discrete torsion free group with a generating set $S$. Let $l$ be the word length function given by $S$. Let $F_n=\{s\in\Gamma| l(s)\leq n\}$.

Assume that $\Lambda$ is a subset of $\Gamma$ such that $$\limsup_{n\to\infty} \frac{|\Lambda\cap F_n|}{|F_n|}>0.$$

Question: For every positive integer $k$, do there exist $b$ and $a$ in $\Gamma$ such that $\{b^{j}a\}_{j=0}^{k-1}\subseteq\Lambda$?

When $\{b^{j}a\}_{j=0}^{k-1}$has $k$ distinct elements, we call it a **left arithmetic progression of length $k$** in $\Gamma$.

Remark: One would avoid the trivial case that $b=e_\Gamma$. In fact, let $\Gamma=\mathbb{Z}$ with $S=\{1\}$, the answer to the above question is affirmative by Szemeredi's theorem, which says that a subset of $\mathbb{Z}$ with positive upper density contains arbitrarily long arithmetic progressions.