Asymmetric infinite sumsets in large sets of integers

We show that for any set A⊂ N with positive upper density and any ,m ∈ N , there exist an infinite set B⊂ N and some t∈ N so that ₁ + b₂ b₁,b₂∈ B\ and\ b₁<b₂ \+t ⊂ A, verifying a conjecture of Kra, Moreira, Richter and Robertson. We also consider the patterns ₁ + b₂ b₁,b₂∈ B\ and\ b₁ ≤ b₂ \ , for infinite B⊂ N and prove that any set A⊂ N with lower density \!d(A)>1/2 contains such configurations up to a shift. We show that the value $1/2$ is optimal and obtain analogous results for values of upper density and when no shift is allowed.