Bohr obstructions to recurrence along Hardy-field sequences

Abstract

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 $R_{u}(E):=\{n\in\mathbb N:E\cap(E-u(n))\neq\varnothing\}$. 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 $\bigcap_{i=1}^k R_{[Mf_i]}(E)$ is not thick. In particular, for some Bohr set $E$, the set $R_{[t^{3/2}]}(E)\cap R_{[\sqrt{2}t^{3/2}+t]}(E)$ is piecewise syndetic but not thick. We also prove that, if for some $\lambda_1,\dots,\lambda_k\in\mathbb{R}$ we have $$\inf_{x\geq 1}\left\|\sum_{i}\lambda_if_i(x)\right\|_{\mathbb{T}}>\frac{1}{2} \sum_i|\lambda_i|,$$ then $\bigcap_i R_{[f_i]}(E)=\varnothing$ 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 $\sup_{x\in\mathbb{R}}|\rho(x)-x|<\infty$.