Separating Bohr denseness from measurable recurrence

We prove that there is a set of integers $A$ having positive upper Banach density whose difference set $A-A:=\{a-b:a,b\in A\}$ does not contain a Bohr neighborhood of any integer, answering a question asked by Bergelson, Hegyv'ari, Ruzsa, and the author, in various combinations. In the language of dynamical systems, this result shows that there is a set of integers $S$ which is dense in the Bohr topology of $\mathbb{Z}$ and which is not a set of measurable recurrence. Our proof yields the following stronger result: if $S\subseteq \mathbb{Z}$ is dense in the Bohr topology of $\mathbb{Z}$, then there is a set $S^{\mathrm{\prime }}\subseteq S$ such that $S^{\mathrm{\prime }}$ is dense in the Bohr topology of $\mathbb{Z}$ and for all $m\in \mathbb{Z},$ the set $(S^{\mathrm{\prime }}-m)\setminus \{0\}$ is not a set of measurable recurrence.