On Erd\H{o}s and S\'ark\"ozy's sequences with Property P

TL;DR

This paper constructs an infinite set of positive integers with Property P, improving the lower bound on its counting function compared to previous examples by Erdős and Sárközy.

Contribution

The authors explicitly construct an infinite set with Property P that has a significantly improved lower bound on its counting function.

Findings

Constructed an infinite set with Property P.

Achieved a lower bound on the counting function involving iterated logarithms.

Improved upon Erdős and Sárközy's earlier bounds.

Abstract

A sequence $A$ of positive integers having the property that no element $a_i \in A$ divides the sum $a_j+a_k$ of two larger elements is said to have `Property P'. We construct an infinite set $S\subset \mathbb{N}$ having Property P with counting function $S(x)\gg\frac{\sqrt{x}}{\sqrt{\log x}(\log\log x)^2(\log \log \log x)^2}$. This improves on an example given by Erd\H{o}s and S\'ark\"ozy with a lower bound on the counting function of order $\frac{\sqrt{x}}{\log x}$.