On sumsets containing a perfect square

TL;DR

This paper proves that the sumset of two large subsets of integers up to N always contains a perfect square, establishing the optimal constant for the minimum sizes needed.

Contribution

It demonstrates the precise threshold for subset sizes ensuring sumsets contain perfect squares, and proves the constant is optimal.

Findings

Sumset contains a perfect square if subsets are large enough

The threshold constant 3/8 is proven to be optimal

Provides a sharp boundary for sumset properties

Abstract

We show $A+B$ contains a perfect square if $A,B \subseteq \{1,\dots,N\}$ have $|A|,|B| \ge (\frac{3}{8}+\epsilon)N$. The constant $\frac{3}{8}$ is optimal.