On Infinite Sum-free Sets of Natural Numbers

AbstractA subset of the natural numbers isk-sum-free if it contains no solutions of the equationx1+…+xk=y, and stronglyk-sum-free when it is ℓ-sum-free for every ℓ=2,…,k. It is shown that everyk-sum-free set with upper density larger than 1/(k+1) is a subset of a periodick-sum-free set and that eachk-sum-free set with upper density larger than 2/(k+3) is subset of ak-sum-free arithmetic progression. In particular, nok-sum-free set has upper density larger than 1/ρ1(k), whereρ1(k)=min{i:i∤k−1}, as conjectured by Calkin and Erdős. Similar problems are studied also for stronglyk-sum-free sets