On exponential type sequences

Abstract

For a sequence A of positive integers, denote by P(A) the set of all sums of distinct terms taken from A. For any integer p>1 and any sequence At={a1≤⋯≤at} of positive integers (not necessarily distinct), let SpAt={piaj:i=0,1,⋯;j=1,2,…,t}. In 2016, Chen, Fang and Hegyvári proved that the sequence P(SpAt) has positive lower asymptotic density if t≥p−1, and the sequence P(SpAt) has asymptotic density zero if 2t<p. In this paper, we further consider the lower asymptotic density and completeness property of SpAt.