Convex sequences may have thin additive bases

TL;DR

This paper constructs large sets with sum sets containing long convex sequences, addressing a question about the structure of additive bases in combinatorics.

Contribution

It provides a method to build arbitrarily large sets whose sum sets include long convex sequences, answering a previously open question.

Findings

Sum sets of large sets can contain convex sequences of quadratic size.

Constructed sets can be arbitrarily large with specified convex sequence length.

Answers a question posed by Hegarty about additive bases.

Abstract

For a fixed $c > 0$ we construct an arbitrarily large set $B$ of size $n$ such that its sum set $B+B$ contains a convex sequence of size $cn^2$, answering a question of Hegarty.