Counterexamples to generalizations of the Erdos B+B+t problem

Following their resolution of the Erdos 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 > 0, a set A N with multiplicative upper Banach density at least 1 - such that A does not contain any dilated product set \b₁b₂t: 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 - such that A does not contain any polynomial configuration \b₁² + 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.