Density thresholds for restricted asymmetric sumsets

Determine whether the density bounds in Proposition 7.1 are sufficient to guarantee shifted and unshifted restricted asymmetric sumsets: for distinct natural numbers [?]ell,m with [?]ell>m, prove or disprove that d(A)>[?]ell/([?]ell+m) implies the existence of an infinite B[0m[?]subset of [?]N and t[0m[?]in [?]N such that {mb1+[?]ellb2:b1,b2[0m[?]in B, b1[0m[?]ne b2} + t[0m[?]subseteq A, and that d(A)>1-m/([?]ell+m)^2 implies the existence of an infinite B[0m[?]subset of [?]N such that {mb1+[?]ellb2:b1,b2[0m[?]in B, b1[0m[?]ne b2}[0m[?]subseteq A.

Background

Section 7 studies asymmetric infinite sumsets in which the diagonal is excluded, namely patterns of the form {mb1+ell b2:b1,b2 in B and b1 != b2}. Earlier results show that positive upper density alone does not ensure such configurations when ell != m. Proposition 7.1 constructs counterexamples at two explicit density levels, suggesting that these levels may be optimal, but the paper does not establish sufficiency above them.

Question 7.3 asks whether the two counterexample thresholds are sharp for shifted and unshifted restricted patterns when ell>m. A positive answer would provide density characterizations analogous to those proved earlier for patterns allowing b1=b2.