Let $A$ be a subset of ${\mathbb N}$ with positive upper-Banach density, and for each integer $k\geq3$, define $R_k=R_k(A)$ to be the smallest positive integer $r$ such that $A$ contains a length $k$ arithmetic progression

$$ \{a, a+r, a+2r, \dots, a+(k-1)r\}. $$

Thus, the finiteness of $R_k$ is Szemeredi's theorem.

What, if anything, is known about how $R_k$ grows? More precisely, the question I am most interested in, without any luck so far, is the following:

Does there exist an example of a set $A$ for which $R_k$ grows with $k$, with, if possible, a lower bound on $R_k$? This lower bound may depend on $A$, but I am hoping for an explicit dependence on $k$.

If there are results in the other direction -- upper bounds on $R_k$ -- I would be interested to hear about those as well. Such a result could be considered a quantitative strengthening of Szemeredi's theorem, so perhaps this is asking for a lot.