Universal density thresholds for restricted asymmetric sumsets

Determine whether every set A[0m[?]subset of [?]N with d(A)>1/2 contains, up to a shift, an infinite restricted asymmetric sumset {mb1+[?]ellb2:b1,b2[0m[?]in B, b1[0m[?]ne b2}, and whether every such set with d(A)>1-1/(2([?]ell+m)) contains an unshifted restricted asymmetric sumset.

Background

The lower-density results earlier in the paper establish sharp thresholds for asymmetric sumsets when the diagonal is included. Proposition 7.4 gives analogous counterexamples at lower density 1/2 for shifted patterns and 1-1/(2(ell+m)) for unshifted patterns when the diagonal is excluded.

Question 7.5 asks whether these lower-density thresholds remain sufficient for restricted asymmetric sumsets, independently of the relative sizes of ell and m. The question is separate from Question 7.3 because it concerns lower natural density rather than upper natural density.

References

Question 7.5. Let ℓ, m ∈ N be distinct and let A ⊂ N. (i) If d(A) > 1/2, does there exist an infinite set B ⊂ N and some t ∈ N such that {mb1 + ℓb2 : b1, b2 ∈ B and b1 6 = b2} + t ⊂ A? (ii) If d(A) > 1 − 1/(2(ℓ + m)), does there exist an infinite set B ⊂ N such that {mb1 + ℓb2 : b1, b2 ∈ B and b1 6 = b2} ⊂ A?

— Asymmetric infinite sumsets in large sets of integers  (2502.03112 - Kousek, 5 Feb 2025) in Question 7.5, Section 7, p. 32