Asymmetric infinite sumsets in large sets of integers

TL;DR

This paper proves that large sets of natural numbers contain complex asymmetric sumset patterns, confirming conjectures and establishing density thresholds for their occurrence with and without shifts.

Contribution

It establishes the existence of infinite asymmetric sumsets within large sets, verifying a conjecture and determining optimal density thresholds for pattern containment.

Findings

Existence of infinite asymmetric sumsets in sets with positive upper density.

Sets with lower density > 1/2 contain specific sumset configurations up to a shift.

Optimal density threshold of 1/2 for pattern occurrence without shifts.

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.