Shifted sumsets in sets of positive density

DefinitionErdős Problem 656 (shifted restricted sumsets in dense sets)#✓fully proved

The statement that for every set \(A\) of natural numbers with positive upper density, that is, with \(\limsup_{N \to \infty} |A \cap \{0, \dots, N - 1\}| / N > 0\), there are an infinite set \(B \subseteq A\) and an integer \(t\) such that \(b_1 + b_2 + t \in A\) for all distinct \(b_1, b_2 \in B\).

def restrictedSumsetShiftInDenseSet : Prop :=
  ∀ A : Set ℕ, 0 < A.upperDensity →
    ∃ B ⊆ A, B.Infinite ∧ ∃ t : ℤ, ∀ b₁ ∈ B, ∀ b₂ ∈ B, b₁ ≠ b₂ →
      ∃ a ∈ A, (a : ℤ) = b₁ + b₂ + t

Uses 1 · Used by 1

Statement usesUpper density

Used byErdős Problem 656 is true

TheoremErdős Problem 656 is true#●ready to prove

For every set \(A\) of natural numbers with positive upper density, that is, with \(\limsup_{N \to \infty} |A \cap \{0, \dots, N - 1\}| / N > 0\), there are an infinite set \(B \subseteq A\) and an integer \(t\) such that \(b_1 + b_2 + t \in A\) for all distinct \(b_1, b_2 \in B\).

This is the theorem of Kra, Moreira, Richter and Robertson.

public theorem_wanted restrictedSumsetShiftInDenseSet_holds : restrictedSumsetShiftInDenseSet

Sources

• T. F. Bloom, Erdős Problems, Problem #656, https://www.erdosproblems.com/656 (status PROVED); resolution: B. Kra, J. Moreira, F. K. Richter and D. Robertson, A proof of Erdős's B + B + t conjecture, Commun. Am. Math. Soc. (2024), 480-494, arXiv:2206.12377, abstract: every set A of natural numbers with positive upper density can be shifted to contain the restricted sumset {b₁ + b₂ : b₁, b₂ ∈ B, b₁ ≠ b₂} for some infinite set B ⊂ A, https://arxiv.org/abs/2206.12377.

Uses 1

Statement usesErdős Problem 656 (shifted restricted sumsets in dense sets)

1 fully proved1 ready to prove

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