Counterexamples to generalizations of the Erd{o}s $B+B+t$ problem

Following their resolution of the Erd{o}s $B+B+t$ problem, Kra Moreira, Richter, and Robertson posed a number of questions and conjectures related to infinite configurations in positive density subsets of the integers and other amenable groups. We give a negative answer to several of these questions and conjectures by producing families of counterexamples based on a construction of Ernst Straus. Included among our counterexamples, we exhibit, for any ε > 0, a set A ⊆ N with multiplicative upper Banach density at least 1 - ε such that A does not contain any dilated product set ₁b₂t : b₁, b₂ ∈ B, b₁ ≠ b₂\ for an infinite set B ⊆ N and t ∈ Q>0. We also prove the existence of a set A ⊆ N with additive upper Banach density at least 1 - ε such that A does not contain any polynomial configuration ₁² + b₂ + t : b₁, b₂ ∈ B, b₁ < b₂\ for an infinite set B ⊆ N and t ∈ Z. Counterexamples to some closely related problems are also discussed.