If S is an arbitrary sequence of
positive integers, define P(S) to be the set of all integers which are representable as a
sum of distinct terms of S. Call a sequence S complete if P(S) contains
all sufficiently large integers, and subcomplete if P(S) contains an infinite
arithmetic progression. We will prove the following theorem: Let n-th term of the
integer sequence S have the form f(n) + O(nα), where f is a polynomial
and where 0 ≦ α < 1; then S is subcomplete. We further show that S is
complete if, in addition, for every prime p there are infinitely many terms of S
not divisible by p. (We call any sequence satisfying this last property an
R-sequence.) We will then extend these results to considerably more general
sequences.
