Remarks on Steinhaus' property and ratio sets of sets of positive integers

This paper is closely related to an earlier paper of the author and W. Narkiewicz (cf. 7]) and to some papers concerning ratio sets of positive integers (cf. 4], 5], 12], 13], 14]). The paper contains some new results completing results of the mentioned papers. Among other things a characterization of the Steinhaus property of sets of positive integers is given here by using the concept of ratio sets of positive integers.