On sets of integers which are both sum-free and product-free

TL;DR

This paper investigates sets of positive integers that are both sum-free and product-free, establishing upper bounds on their density and exploring their periodic structures, with the results showing the maximum possible densities are strictly below 1/2.

Contribution

It proves the upper density of sum- and product-free sets is less than 1/2 and identifies the maximal density for periodic such sets, providing tight bounds.

Findings

Upper density of such sets is strictly less than 1/2

Maximum density for periodic sets is characterized

Bounds are shown to be optimal

Abstract

We consider sets of positive integers containing no sum of two elements in the set and also no product of two elements. We show that the upper density of such a set is strictly smaller than 1/2 and that this is best possible. Further, we also find the maximal order for the density of such sets that are also periodic modulo some positive integer.