Counterexamples to integer-coefficient criteria for recurrence along functions from a Hardy field

We give negative answers to two questions of Bergelson, Moreira, and Richter concerning recurrence along functions from a Hardy field. For the pair $f_1(t)=t^{3/2}$ and $f_2(t)=\lambda t^{3/2}+t$, where $\lambda\in\mathbb R\setminus\mathbb Q$, singled out in their integer-coefficient derivative-span question, we prove that every $F\in\nablaz(f_1,f_2)$ satisfies $\lim_{t\to\infty}|F(t)|\in\{0,\infty\}$. Nevertheless, there is a set $E\subset\mathbb N$ of positive natural density such that $R_{f_1}(E)\cap R_{f_2}(E)$ is piecewise syndetic and not thick. Thus the proposed integer-coefficient replacement does not imply thickness. We further show that, even under the same full integer derivative-span condition, the common return-time set may be empty. This stronger obstruction also gives a negative answer to their question asking whether the recurrence conclusion of Theorem A follows from joint intersectivity of the integer polynomials in $\operatorname{poly}(f_1,\ldots,f_k)$. The constructions use elementary Bohr sets.