Counterexamples to Generalizations of the Erdős B+B+t Problem

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 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 {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 − ε 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.

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BibTeX RIS

BibTeX

@article{ackelsberg2026counterexamples,
  title = {{Counterexamples to Generalizations of the Erdős $B+B+t$ Problem}},
  author = {Ethan Ackelsberg},
  journal = {The Electronic Journal of Combinatorics},
  year = {2026},
  month = sep,
  doi = {10.37236/14787},
  url = {https://doi.org/10.37236/14787},
  note = {CC BY},
}

RIS

TY  - JOUR
TI  - Counterexamples to Generalizations of the Erdős B+B+t Problem
AU  - Ethan Ackelsberg
PY  - 2026
DA  - 2026/09/25
JO  - The Electronic Journal of Combinatorics
DO  - 10.37236/14787
UR  - https://doi.org/10.37236/14787
AB  - 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 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 {b_1b_2t : b_1, b_2 ∈ B, b_1 ≠ b_2} 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 {b_1^2 + b_2 + t : b_1, b_2 ∈ B, b_1 < b_2} for an infinite set B ⊆ ℕ and t ∈ ℤ. Counterexamples to some closely related problems are also discussed.
SN  - 1077-8926
SN  - 1097-1440
ER  -

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