Year: 2026
Type: article
Abstract: Following their resolution of the Erdős $B+B+t$ 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 their questions and conjectures by producing families of counterexamples based on a construction of Ernst Straus.Included among our counterexamples, we exhibit, for any $\varepsilon > 0$, a set $A \subseteq \mathbb{N}$ with multiplicative upper Banach density at least $1 - \varepsilon$ such that $A$ does not contain any dilated product set $\{b_1b_2t : b_1, b_2 \in B, b_1 \ne b_2\}$ for an infinite set $B \subseteq \mathbb{N}$ and $t \in \mathbb{Q}_{>0}$. We also prove the existence of a set $A \subseteq \mathbb{N}$ with additive upper Banach density at least $1 - \varepsilon$ such that $A$ does not contain any polynomial configuration $\{b_1^2 + b_2 + t : b_1, b_2 \in B, b_1 < b_2\}$ for an infinite set $B \subseteq \mathbb{N}$ and $t \in \mathbb{Z}$. Counterexamples to some closely related problems are also discussed.
DOI: 10.37236/14787
Author: Ethan Ackelsberg
Institution: École Polytechnique Fédérale de Lausanne
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4396243886