A proof of two Erdos' conjectures on restricted addition and further results

Erdos was interested in restricted addition and asked many questions on the subject. More than twenty years ago, the two following conjectures were again stated in an important book of problems and questions by Erdos and Graham [11]: given a set of non negative integers, script A sign, if there exists an integer h such that the upper density d̄(hscript A sign) is strictly positive then d̄(h × script A sign) > 0; if a set of non negative integers script A sign is an additive basis of order r then the lower density d̄(r × script A sign) is strictly positive. In this paper, we introduce a new method which allows us to settle both of these conjectures. In addition, we raise some further questions which are partially solved.