Some Remarks on a Theorem of Erdös concerning Asymptotic Density

Due to the asymmetry introduced by the hypothesis (2), the above theorem leaves open the case where ß >ct. Actually, the entire theorem as it stands holds without hypothesis (3). This results from the fact that Erdös' proof of the theorem given in [l ] uses only the condition a>ß/2. In the contrary case, 2a=/3, if we let ai be any fixed integer of A, the set {ö+cti}, b^B, has asymptotic density ß~\ta+ß/2; and the theorem follows. In this note it is proposed to give (using only hypotheses (1) and (2)) a short proof of (4) which does not however yield the more precise information concerning the sets of (5). This proof is based upon the (a, ß) hypothesis for Schnirelman density, proved in [2], [3]. If either a or ß equals 0, (4) follows trivially. Hence we may assume that both a and ß are not 0. Since a is the asymptotic density of A, given any e, a>e>0, we can find a largest integer m such that