Let $A\subseteq [0,\infty)$ be a set containing infinite arithmetic progressions of ANY gap, that is, for any $d>0$, there is $t>0$ such that $t+kd\in A$ for all $k\in \mathbb N$.
Molter and Yavicoli (2016) showed that there exists such set $A$ which is $F_\sigma$ and has zero Hausdorff dimension. Their set, however, is dense in some $[M,\infty)$ (in particular, it cannot be closed).
My question is: does there exist a CLOSED set of zero Hausdorff dimension with the above property?
I attempted to show that if $A$ contains infinite arithmetic progressions of any gap, then $A$ must be dense somewhere, so a closed $A$ contains an interval. However, I was wrong, as the following post gives me a counterexample:
Nevertheless, I can show that if $A\subseteq [0,\infty)$ contains a translated copy of any sequence $\alpha_i$ increasing to infinity, then $A$ must be dense in some $[M,\infty)$.
