Moment conditions for the asymptotic completeness of iid gap sequences

Abstract

We study conditions under which integer sequences with independent, identically distributed gaps are asymptotically $k$-complete, meaning that every sufficiently large integer can be represented as the sum of exactly $k$ distinct elements of the sequence, or equivalently whether $k$-fold sumsets with distinct entries from such sequences generate all sufficiently large integers. Prior results established asymptotic completeness under strong conditions on the gap distribution involving the moment generating function. Leveraging renewal theory our main result shows that asymptotic $2$-completeness holds almost surely under the much weaker assumption of a finite second moment. Furthermore, using Schnirelmann densities and Mann's theorem we show weak asymptotic $k$-completeness under only a finite first moment condition, albeit with an upper bound on the first moment.