On sets of integers which are both sum-free and product-free
  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. 
Par Kurlberg, Jeffrey C. Lagarias, Carl Pomerance
math.NT