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 $\{m{b}_1+\ell b_2:b_1,b_2\in B\text{and}{b}_1<b_2\}+t\subset A,$ verifying a conjecture of Kra, Moreira, Richter and Robertson. We also consider the patterns $\{m{b}_1+\ell b_2:b_1,b_2\in B\text{and}{b}_1\le b_2\}$, 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.

34 pages

References in corpus (2)

• Infinite Sumsets in Sets with Positive Density
• Finding product sets in some classes of amenable groups