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

TL;DR

This paper constructs counterexamples to several conjectures related to the Erdős $B+B+t$ problem, demonstrating that high-density sets can lack certain infinite configurations, thus challenging previous assumptions.

Contribution

It provides explicit counterexamples showing that sets with high density can avoid specific infinite polynomial and multiplicative configurations, answering open questions negatively.

Findings

Existence of high-density sets avoiding certain multiplicative configurations.

Existence of high-density sets avoiding specific polynomial configurations.

Counterexamples challenge previous conjectures about configurations in dense sets.

Abstract

Following their resolution of the Erd\H{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 $\varepsilon > 0$, 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.