Special cases and equivalent forms of Katznelson’s problem on recurrence

We make the following three observations regarding a question popularized by Katznelson: is every subset of $${\mathbb {Z}}$$ which set Bohr recurrence also topological recurrence? (i) If G countable abelian group and $$E\subseteq G$$ an $$I_0$$ set, then $$E-E$$ recurrence. In particular $$\{2^n-2^m : n,m\in {\mathbb {N}}\}$$ (ii) Let {Z}}^{\omega }$$ be direct sum countably many copies with standard basis E. $$(E-E)-(E-E)$$ recurrence, (iii) Fix prime p let {F}}_p^\omega $$ {Z}}/p{\mathbb $$({\mathbf {e}}_i)_{i\in {N}}}$$ . for p-uniform hypergraph vertex {N}}$$ edge $${\mathcal {F}}$$ having infinite chromatic number, Cayley graph on $$\mathbb F_p^\omega determined $$\{\sum _{i\in F}{\mathbf {e}}_i:F\in {\mathcal {F}}\}$$ has