Upper-density thresholds for unrestricted asymmetric sumsets

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

Background

The unrestricted pattern mB+ell B allows both equal and unequal indices. Proposition 7.6 constructs sets at the same upper-density thresholds that obstruct these patterns, while the preceding theorems establish sufficiency for the diagonal-inclusive ordered pattern b1<=b2. The missing issue is whether those thresholds also suffice when no ordering restriction is imposed.

Question 7.8 focuses on the case m>ell and asks for sharp upper-density criteria for shifted and unshifted unrestricted asymmetric sumsets.

References

Question 7.8. Let ℓ, m ∈ N be distinct with m > ℓ and let k = m/ℓ. Let also A ⊂ N. (i) If d(A) > (k + 1)/(k + 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/ (ℓ(k + 1)(k + 2)), 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 7.8, Section 7, p. 33