Bohr obstructions to recurrence along Hardy-field sequences

We construct Bohr obstructions to multiple recurrence along rounded Hardy-field sequences, showing that the real derivative-span criterion of Bergelson, Moreira, and Richter is essentially sharp and answering two of their questions. For $E\subseteq \mathbb{N}$ and $u:\mathbb{N}\to \mathbb{Z}$, set $R_u(E):=\{n\in \mathbb{N}:E\cap (E-u(n))\mathrm{\ne }\mathrm{\varnothing }\}$. We prove that, if $f_1,\dots ,f_k$ are functions of polynomial growth from a Hardy field and some real linear combination of $f_1,\dots ,f_k$ and their derivatives has a nonzero finite limit, then there exist $M\in \mathbb{N}$ and a basic Bohr set $E\subseteq \mathbb{N}$ such that ${\bigcap }_{i=1}^k{R}_{[M{f}_i]}(E)$ is not thick. In particular, for some Bohr set $E$, the set $R_{[t^{3\mathrm{/}2}]}(E)\cap R_{[\sqrt{2}{t}^{3\mathrm{/}2}+t]}(E)$ is piecewise syndetic but not thick. We also prove that, if for some ${\lambda }_1,\dots ,{\lambda }_k\in \mathbb{R}$ we have $$\underset{x\ge 1}{\inf }{\parallel \sum _i{\lambda }_i{f}_i(x)\parallel }_{\mathbb{T}}>\frac{1}{2}\sum _i\mathrm{\mid }{\lambda }_i\mathrm{\mid },$$ then ${\bigcap }_i{R}_{[f_i]}(E)=\mathrm{\varnothing }$ for some basic Bohr set $E$. More generally, our results apply with $[\cdot ]$ replaced by any rounding function $\rho :\mathbb{R}\to \mathbb{Z}$ satisfying ${\sup }_{x\in \mathbb{R}}\mathrm{\mid }\rho (x)-x\mathrm{\mid }<\mathrm{\infty }$.