Lower-density thresholds for unrestricted asymmetric sumsets

Determine whether every set A[0m[?]subset of [?]N with d(A)>1/2 contains an unrestricted asymmetric sumset {mb1+[?]ellb2:b1,b2[0m[?]in B} up to a shift, and whether every set with d(A)>1-1/(2([?]ell+m)) contains such a sumset without a shift, for distinct natural numbers [?]ell and m.

Background

Proposition 7.7 supplies counterexamples at lower-density levels 1/2 and 1-1/(2(ell+m)) for unrestricted asymmetric sumsets. These values coincide with the optimal lower-density thresholds obtained earlier for diagonal-inclusive patterns, but the unrestricted case remains unresolved for distinct coefficients.

Question 7.9 asks whether the lower-density bounds are sufficient for unrestricted shifted and unshifted patterns, thereby extending the lower-density theory beyond the already treated ordered configurations.

References

Question 7.9. Let ℓ, m ∈ N be distinct and 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} + 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} ⊂ A?

— Asymmetric infinite sumsets in large sets of integers  (2502.03112 - Kousek, 5 Feb 2025) in Question 7.9, Section 7, p. 33