Moment conditions for the asymptotic completeness of iid gap sequences
Vahram Asatryan, Erik Babasyan, Sevak Mkrtchyan
TL;DR
The paper investigates asymptotic completeness properties of random weight sequences generated by i.i.d. gaps, and shows that almost-sure asymptotic $2$-completeness holds under the weaker condition $\mathbb{E}[X^2]<\infty$ via renewal-theoretic intersection arguments. It further establishes weak asymptotic $k$-completeness under the finite first moment condition $\mathbb{E}[X]<\infty$ by leveraging Schnirelmann density and Mann's theorem, with a special strengthening to asymptotic $2$-completeness when $k=2$. The work links delayed renewal process intersections to additive representation in random sequences and clarifies how moment conditions govern the transition from weak to strong forms of asymptotic completeness. These results relax previous moment-generating-function requirements and illuminate the role of renewal theory in random sumset problems, offering a bridge between probabilistic renewal techniques and additive combinatorics.
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.