Asymmetric infinite sumsets in large sets of integers

Abstract

We show that for any set $A \subset \mathbb{N}$ with positive upper density and any $\ell,m \in \mathbb{N}$, there exist an infinite set $B\subset \mathbb{N}$ and some $t\in \mathbb{N}$ so that $\{mb_1 + \ell b_2 \colon b_1,b_2\in B\ \text{and}\ b_1<b_2 \}+t \subset A,$ verifying a conjecture of Kra, Moreira, Richter and Robertson. We also consider the patterns $\{mb_1 + \ell b_2 \colon b_1,b_2\in B\ \text{and}\ b_1 \leq b_2 \}$, for infinite $B\subset \mathbb{N}$ and prove that any set $A\subset \mathbb{N}$ with lower density $\underline{d}(A)>1/2$ 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.